Hyperbola A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant. b 311, - 311. Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject read Perimeter of an Ellipse for more details. Find the standard form of the equation for a hyperbola with vertices at (0, -8) and (0, 8) and asymptote y 2x y 2 x. This calculator will find either the equation of the hyperbola from the given parameters or the center, foci, vertices, co-vertices, (semi)major axis length, (semi)minor axis length,. Use this form to determine the values used to find vertices and asymptotes of the hyperbola. P F1-PF2 &39;2a&39;. Since the vertices are a 4 units to either side, then they are at the points (7, 2) and at (1, 2). Hyperbola Calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The two given points are the foci of the hyperbola, and the midpoint of the segment joining the foci is the center of the hyperbola. Two fixed points located inside each curve of a hyperbola that are used in the curve&39;s formal definition. Solution to Example 3. A hyperbola (plural "hyperbolas"; Gray 1997, p. Note that this places them inside the hyperbola. For any point on any of the branches, the absolute difference between the point from foci is constant and equals 2a, where. Find an appropriate equation. Major Axis The length of the major axis of the hyperbola is. Free Hyperbola Foci (Focus Points) calculator - Calculate hyperbola focus points given equation step-by-step. The object enters along a path approximated by the line y x2 y x 2 and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. Improve this question. The line segment between the two vertices is known as the transverse or major axis. Free Ellipse Foci (Focus Points) calculator - Calculate ellipse focus points given equation step-by-step. Semi Transverse Axis of Hyperbola - (Measured in Meter) - Semi Transverse Axis of Hyperbola is half of the distance between the vertices of the Hyperbola. The points F1 and F2 are called the foci of the hyperbola. Hyperbola Find Equation Gvien Focus, Transverse Axis Length. Free Ellipse Foci (Focus Points) calculator - Calculate ellipse focus points given equation step-by-step. With this definition, we may now define a conic in terms of the directrix, the eccentricity and the angle Thus, each conic may be written as a polar equation, an equation written in terms of and. Each of the fixed points is called a focus of the hyperbola. standard form of a hyperbola that opens up and down, it is (y - k) 2 a 2 - (x - h) 2 b 2 1. The hyperbola foci formula is the same for vertical and horizontal hyperbolas and looks like the Pythagorean Theorem eqa2 b2 c2 eq where c represents the focal distance (the distance. The line segment of length 2b perpendicular to the transverse axis whose midpoint is the center is the conjugate axis of the hyperbola. Solution Step 1 Find the center of the hyperbola. A hyperbola is the set of all points in the plane the difference of whose distances from two fixed points is some constant. In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. The center is midway between the two foci, so the center must be at (h, k) (1, 0). Stan at (2. The two given points are the foci of the hyperbola, and the midpoint of the segment joining the foci is the center of the hyperbola. See Figure 8. c is the distance between the focus (- 4, 3) and the center (2, 3). Free Hyperbola Center calculator - Calculate hyperbola center given equation step-by-step. a semi-major axis of the hyperbola. 5, 0), and (-7. This calculator will find either the equation of the hyperbola from the given parameters or the center, foci, vertices, co-vertices, (semi)major axis length, (semi)minor axis length, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, asymptotes, x-intercepts, y-intercepts, domain, and. Hyperbolae were discovered by Menaechmus in his investigations of the problem of doubling the cube, but were then called sections of obtuse cones. The center is at (h, k). Free Hyperbola Calculator - Given a hyperbola equation, this calculates Equation of the asymptotes. This special property is the foundation for Loran. An equation of this hyperbola can be found by using the. To simplify the equation of the ellipse, we letc2 a2 b2. Solve ellipses step by step. Explore math with our beautiful, free online graphing calculator. If the slope is 0 0, the graph is horizontal. The center is the midpoint between the two vertices, so we have. Google Classroom. Let us check through a few important terms relating to the different parameters of a hyperbola. A hyperbola is a type of conic section that looks somewhat like a letter x. b 52 42 9 3. The foci of an ellipse equation calculator is showing the foci of an ellipse. There are also two lines on each graph. Determine the center, vertices, and foci of the hyperbola with the equation 9x 2 4y 2 36. Hyperbola - With Asymptotes Desmos. Hyperbola Definition. y2 - 4x2 - 4y 24x 48. Explore math with our beautiful, free online graphing calculator. Asymptote 1 . The hyperbola has a very important distance property that helps locate the ship. Let us check through a few important terms relating to the different parameters of a hyperbola. The given equation is that of hyperbola with a vertical transverse axis. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. they will be (7,-1) and (-1,-1). Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. You might need Calculator. The latus rectum is a line that runs parallel to the conic's directrix and passes through its foci. Popular Problems Algebra Find the Hyperbola Center (5,6), Focus (-5,6), Vertex (4,6) (5,6) , (4,6) , (-5,6) (5, 6) , (4, 6) , (- 5, 6) There are two general equations for a hyperbola. At the beginning of the video he shows you the ellipse because he wanted you to see that f sqrt (a2 - b2) is an equation that applies to the ellipse and then then after that he starts to talk about the hyperbola and how the equation is f sqrt (a2 b2), which shows the relationship you are asking of between the ellipse and the hyperbola. Latus Rectum of Hyperbola. Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. b semi-minor axis of the hyperbola. The distance between the foci is 2c. The center is at (h,k) (-sqrt(2), 2). foci x2-y21. Explore math with our beautiful, free online graphing calculator. Remember, x and. 5, 0). Explore math with our beautiful, free online graphing calculator. STEP 2 Evaluate Formula. d distance from center to any one of the focii of the hyperbola. The hyperbola opens left and right, because the x term appears first in the standard form. For more math formulas, check out our Formula Dossier. standard form of a hyperbola that opens up and down, it is (y - k) 2 a 2 - (x - h) 2 b 2 1. Just as with ellipses, writing the equation for a hyperbola in standard form allows us to calculate the key features its center, vertices, co-vertices, foci, asymptotes, and the lengths and positions of the transverse and conjugate axes. In analytic geometry, a hyperbola is a conic section formed by intersecting a right circular cone with a plane at an angle such that both halves of the cone are intersected. Use this form to determine the values used to find vertices and asymptotes of the hyperbola. The length of the latus rectum of the hyperbola having the standard equation of x 2 a 2 - y 2 b 2 1, is 2b 2 a. There are also two lines on each graph. Related Symbolab blog posts. Equation of a hyperbola from features. For more math formulas, check out our Formula Dossier. Explore math with our beautiful, free online graphing calculator. More on hyperbolas. Hyperbolas, An Introduction - Graphing Example. Popular Problems Algebra Find the Hyperbola Center (5,6), Focus (-5,6), Vertex (4,6) (5,6) , (4,6) , (-5,6) (5, 6) , (4, 6) , (- 5, 6) There are two general equations for a hyperbola. a semi-major axis of the hyperbola. The variable h h represents the x-offset from the origin, k k represents the y-offset from origin, a a. Place the thumbtacks in the cardboard to form the foci of the ellipse. A given point of a parable is at the same distance from both the focus and the directrix. Filling in the values for a 2 and b 2. Free Hyperbola Foci (Focus Points) calculator - Calculate hyperbola focus points given equation step-by-step. Hyperbola Calculator. Explore math with our beautiful, free online graphing calculator. Conjugate Axis of Hyperbola formula is defined as the line through the center and perpendicular to transverse axis with length of the chord of the circle passing through the foci and touches the Hyperbola at vertex is calculated using Conjugate Axis of Hyperbola 2 Semi Conjugate Axis of Hyperbola. If you're wondering what the latus rectum is or how to find the latus rectum, you've come to the right place. The co vertices in the x direction is The equation of the hyperbola is The foci are at the points (0 , 10) and (0 , 10) Latus rectum coordinate is the value x 0 of the graph at the point y 0 c 10. Hyperbola in Standard Form and Vertices, Co Vertices, Foci, and Asymptotes of a Hyperbola Example 1 Find the center and foci of &92;(x2y28x-4y-440&92;) Solution. Note that this places them inside the hyperbola. The denition of hyperbola involves the difference rather than the sum. calculate the equation of the asymptotes. 2 Answers. Explore math with our beautiful, free online graphing calculator. Horizontal means the right side of the equation is 1, vertical means the right side is -1. It will draw and calculate the area, circumference, and foci for any size ellipse. Sorted by 0. The center is midway between the two foci, so the center must be at (h, k) (1, 0). Thus, every point X X of a hyperbola whose foci are points F F. For more math formulas, check out our Formula Dossier. Step 2 Now click the button Calculate to get the values of a hyperbola. where , b 2 a 2 (e 2 1) Important Terms and Formulas of Hyperbola. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. As a. standard form of a hyperbola that opens up and down, it is (y - k) 2 a 2 - (x - h) 2 b 2 1. Eccentricity . We can calculate the eccentricity of a conic section using values from its equation as in those that. foci asymptotes (x,y) (x,y) ((smaller x-value)) ((arger x-value) (smaller slope) (larger slope) There are 3 steps. Jun 5, 2023 Our latus rectum calculator will obtain the latus rectum of a parabola, hyperbola, or ellipse and their respective endpoints from just a few parameters describing your function. This is going to be a comma 0. Example Graphing a Hyperbola Centered at (0, 0) Given an Equation in Standard Form. Free Parabola Directrix calculator - Calculate parabola directrix given equation step-by-step. Definition Hyperbola. Explore math with our beautiful, free online graphing calculator. The asymptote of the line. If the slope is 0 0, the graph is horizontal. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. It is also the focal chord parallel to the directrix. Learn how to find the center of a hyperbola, and how to calculate the focal points using the hyperbola foci formula. In the figure above. Hyperbola - Horizontal Transverse Axis Desmos. It will draw and calculate the area, circumference, and foci for any size ellipse. 45) is a conic section defined as the locus of all points in the plane the difference of whose distances and from two fixed points (the foci and) separated by a distance is a given positive constant , (1) (Hilbert and Cohn-Vossen 1999, p. semi. Send us Feedback. Hyperbola - With Asymptotes Desmos. e 1, the conic is a parabola. Explore math with our beautiful, free online graphing calculator. Definition Hyperbola. For more math formulas, check out our Formula Dossier. Conversely, an equation for a hyperbola can be found given its key features. In mathematics this is called unbounded, which means no circle, no matter how large, can enclose the shape. Axis of Hyperbola The line passing through the foci and the center of the hyperbola is the axis of the hyperbola. POLAR EQUATION FOR A CONIC SECTION. Hyperbola is an open curve that has two branches that look mirror image of each other. This calculator will find either the equation of the hyperbola from the given parameters or the center, foci, vertices, co-vertices, (semi)major axis length, (semi)minor axis length, latera recta, length of the latera recta (focal width), focal parameter, eccentricity, linear eccentricity (focal distance), directrices, asymptotes, x-intercepts,. Step 3 Finally, the focus, asymptote, and eccentricity will be displayed in the output field. This Hyperbola is horizontal. When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. 67 21. We will cover those questions (and more) below, paired. If the slope is 0 0, the graph is horizontal. Identify and label the vertices, co-vertices, foci, and asymptotes. This formula uses 2 Variables. The foci of a hyperbola are located at &92;left (&92;frac c 2,0&92;right) &92;text and &92;left (-&92;frac c 2,0&92;right). Click here to view image. (xh)2 a2 (yk)2 b2 1 (x - h) 2 a 2 (y - k) 2 b 2 1. Solved Examples on Hyperbola Calculator. The co vertices in the x direction is The equation of the hyperbola is The foci are at the points (0 , 10) and (0 , 10) Latus rectum coordinate is the value x 0 of the graph at the point y 0 c 10. Example 1 Find the standard form equation of the hyperbola with vertices at (-4,0) and (4,0) and foci at (-6,0) and (6,0). Finding the Equation for a Hyperbola Given the Graph - Example 2. Horizontal hyperbola equation (x h)2 a2 (yk)2 b2 1 (x - h) 2 a 2 - (y - k) 2 b 2 1. a distance from vertices to the center. Since in the pattern the denominators are a 2 and b 2, we can substitute those right into the formula c 2 a 2 b 2. The object enters along a path approximated by the line y x2 y x 2 and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. A hyperbola consists of two curves opening in opposite directions. For a hyperbola &92;(&92;dfracx2a2 - &92;dfracy2b2 1&92;) the x-axis is the axis of hyperbola and has the equation y 0. Interactive online graphing calculator - graph functions, conics, and inequalities free of charge. I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is 2a 2 a, the distance between the two vertices. When the transverse axis is horizontal, the equation of the hyperbola graph calculator will be (xh) 2 a 2 (yk) 2 b 2 1. This calculator will find either the equation of the hyperbola from the given parameters or the center, foci, vertices, co-vertices,. The standard form of a quadratic equation is y ax&178; bx c. Hyperbola Graph. A Circle has an Eccentricity equal to zero, so the Eccentricity shows you how un - circular the given curve is. Drag point &39;a,b&39; or sliders to change shape and point P to change mirror reflections. Hyperbola from Foci. I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is 2a 2 a, the distance between the two vertices. We can find the value of c by using the formula c2 a2 - b2. Hyperbola Formulas. Conversely, an equation for a hyperbola can be found given its key features. A hyperbola centered at the origin has vertices at (7, 0) and foci at (27, 0). x2 a2 y2 c2 a2 1 So, the equation of a hyperbola centered at the origin in standard form is x2 a2 y2 b2 1. Hyperbola opens upward and downward. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Free Hyperbola Eccentricity calculator - Calculate hyperbola eccentricity given equation step-by-step. When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate. Example 9. Since the vertices are a 4 units to either side, then they are at the points (7, 2) and at (1, 2). Question Consider the following. I also see that you know that the slope of the asymptote line of a hyperbola is the ratio &92;dfracba for a simple hyperbola of the form &92;dfracx2a2-&92;dfracy2b21. Foci of hyperbola (ae, 0) (5 &215; 32, 0) (7. Hyperbola from Foci. The equation of a hyperbola is given by (y 2)2 32 (x 3)2 22 1. Explore math with our beautiful, free online graphing calculator. Jun 4, 2020 The co vertices in the x direction is The equation of the hyperbola is The foci are at the points (0 , 10) and (0 , 10) Latus rectum coordinate is the value x 0 of the graph at the point y 0 c 10. If the slope is undefined, the graph is vertical. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. We calculate the distance from the point on the ellipse (x, y) to the two foci, (0, 5) and (0, -5). Free Hyperbola Axis calculator - Calculate hyperbola axis given equation step-by-step. Below are some solved examples on hyperbola calculator general form. Hyperbola Focus F&39; X Coordinate x 0 - . What 2 formulas are used for the Hyperbola Calculator standard form of a hyperbola that opens sideways is (x - h) 2 a 2 - (y - k) 2 b 2 1. A hyperbola centered at the origin has vertices at (7, 0) and foci at (27, 0). The steps for graphing a hyperbola given its equation in general form are outlined in the following example. Google Classroom. laidhub, the reason synonyms
And the latus rectum length is L 2 x 0 2 10. . Hyperbola foci calculator
When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate. These lines are called asymptotes and as the graphs show as we make x. This intersection produces two separate unbounded curves that are mirror images of each other (Figure (PageIndex2)). Our Pre-Calculus Calculator covers a wide range of pre-calculus topics, including functions, equations, inequalities, trigonometry, conic sections, complex numbers, and more. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Log InorSign Up. To use this online calculator for Eccentricity of Hyperbola, enter Semi Conjugate Axis of Hyperbola (b) & Semi Transverse Axis of Hyperbola (a) and hit the calculate button. Find an equation for the hyperbola. Find the standard form of the equation for a hyperbola with vertices at (0, -8) and (0, 8) and asymptote y 2x y 2 x. If the slope is 0 0, the graph is horizontal. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Free Hyperbola Axis calculator - Calculate hyperbola axis given equation step-by-step. Some applications of hyperbolas are given in the exercises. Therefore, you will have the equation of the standard form of hyperbola calculator as c 2 a 2 b 2 b c 2 a 2. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. To calculate Conjugate Axis of Hyperbola, you. The latus rectum and the directrix are perpendicular to the axis of the hyperbola. 14 Graph of the equation xy 1; The red lines indicate the rotated axes. A hyperbola centered at the origin has vertices at (7, 0) and foci at (27, 0). Jul 8, 2021 To name the foci as points in a horizontal hyperbola, you use (h F, v); to name them in a vertical hyperbola, you use (h, v F). Step 2 Because the hyperbola is horizontal and c 10, the foci are located c units to the right and left of the center, where c satisfies c 2 a 2 b 2. Each fixed point is called a focus (plural foci). Given two distinct points F1 and F2 in the plane and a fixed distance d, a hyperbola is the set of all points (x, y) in the plane such that the absolute value of the difference of each of the distances from F1 and F2 to (x, y) is d. The latus rectum and the directrix are perpendicular to the axis of the hyperbola. The other two cones are elliptical and parabolic. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. Our latus rectum calculator will obtain the latus rectum of a parabola, hyperbola, or ellipse and their respective endpoints from just a few parameters describing your function. An ellipse is a curve that is the locus of all points in the plane the sum of whose distances r1 and r2 from two fixed points F1 and F2 (the foci) separated by a distance of 2c is a given positive constant 2a (Hilbert and Cohn-Vossen 1999, p. Free Hyperbola Foci (Focus Points) calculator - Calculate hyperbola focus points given equation step-by-step. We can calculate the eccentricity of a conic section using values from its equation as in those that. For more math formulas, check out our Formula Dossier. 6 sqrt (1 (122) (52)). The center is midway between the two foci, so the center must be at (h, k) (1, 0). The Hyperbolas. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. The distance between each focus and the center is called the focal length of the ellipse. 14 Graph of the equation xy 1; The red lines indicate the rotated axes. When the transverse axis is horizontal, the equation of the hyperbola graph calculator will be (xh) 2 a 2 (yk) 2 b 2 1. Figure 11. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. A hyperbola has two latus rectums. To derive it, use the eccentricity formula e (a - b) a, where a 5 and b 4. Conic Sections. Oct 6, 2021 the equations of the asymptotes are y b ax y b a x. A hyperbola is a conic section that is the set of all points in a plane such that the difference of the distances from two fixed points (foci) is a constant. See Example (PageIndex2) and Example (PageIndex3). Leave a Reply Cancel reply. You might need Calculator. W e CMawdrei MwIiZtbhv RIMndfqiznkidtJex RPrNecUaQlwceuPlUuqsM. I have recently learned about the Hyperbolas and it's properties. y2 a2 x2 b2 1 y 2 a 2 x 2 b 2 1. A given point of a parable is at the same distance from both the focus and the directrix. Log InorSign Up. Hyperbola Creator Desmos. Hint Your hyperbola has equation y2 a2 x2 b2 y2 4 x2 4 1 y 2 a 2 x 2 b 2 y 2 4 x 2 4 1. Question Find the center, transverse axis, vertices, foci, and asymptotes. It is also the focal chord parallel to the directrix. Figure 11. Hyperbola - Horizontal Transverse Axis Desmos. In other words, a hyperbola is a set of all points on the planes, for which the absolute value of the difference between the distances and two fixed points (known as foci of hyperbola) is constant. Find the standard form of the equation for a hyperbola with vertices at (0, -8) and (0, 8) and asymptote y 2x y 2 x. Explore math with our beautiful, free online graphing calculator. My Notebook, the Symbolab way. The following is an example of a hyperbola. Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism,. Hyperbola With Foci. Note the sign difference from an ellipse where it&39;s &92;sqrta2-b2. It no longer matters which has the greater value of a or b. The above figure represents a hyperbola such that P 1 F 2 P 1 F 1 P 2 F 2 P 2 F 1. In the figure above. Finding the Equation for a Hyperbola Given the Graph - Example 1. Solve a hyperbola by finding the x and y intercepts, the coordinates of the foci, and drawing the graph of the equation. This constant equals the distance between the vertices of the hyperbola. Given the hyperbola with the equation (x 2) 2 16 (y 1) 2 9 1, find the coordinates of its center, vertices, and foci. This intersection produces two separate unbounded curves that are mirror images of each other (Figure &92;(&92;PageIndex2&92;)). I also see that you know that the slope of the asymptote line of a hyperbola is the ratio &92;dfracba for a simple hyperbola of the form &92;dfracx2a2-&92;dfracy2b21. Hyperbola Calculator. y 2 b x. Additionally, it draws the parabolas plots as well. , where f, a, and b is the focal length, distance from center to vertices, and distance from center to the co-vertices, respectively, is confusing to me that from where does this formula. Find an equation for the hyperbola. endgroup N. standard form of a hyperbola that opens up and down, it is (y - k) 2 a 2 - (x - h) 2 b 2 1. The Hyperbola in Standard Form. The answer is 35. Graph the following hyperbola and mark its foci &92;(&92; 16 x264 x-9 y290 y-3050&92;). Step 3 We find a2 a2 using the distance between the vertices, 2a 2a. ) You can drag point A, one of the focus points, which will change the shape of the. A conic section with a focus at the origin, eccentricity e, and directrix at x p or y p will have polar equation r ep 1 esin() when the directrix is y p. The equation of the hyperbola is simplest when the centre of the hyperbola is at the origin, and the foci are either on the x-axis or on the y-axis. The center is at (h, k). Math notebooks have been around for hundreds of years. The distance between each focus and the center is called the focal length of the ellipse. Free Hyperbola calculator - Calculate Hyperbola center, axis, foci, vertices, eccentricity and asymptotes step-by-step. 4 Recognize a parabola, ellipse, or hyperbola from its eccentricity value. For a hyperbola &92;(&92;dfracx2a2 - &92;dfracy2b2 1&92;) the x-axis is the axis of hyperbola and has the equation y 0. -1) on a horizontal line, one to the right of (2. The endpoints of the latus rectum of the hyperbola passing through the focus (ae, 0), is (ae, b 2 a), and (ae, -b 2 a). Asymptote 2 . Jul 24, 2016 3) Compare the given focus with the center. 9) Vertices 11) Vertices 13) Center at. Given two distinct points F1 and F2 in the plane and a fixed distance d, a hyperbola is the set of all points (x, y) in the plane such that the absolute value of the difference of each of the distances from F1 and F2 to (x, y) is d. Use the coordinates of the center point (h, k) to plug the values of h and k into the formula. Find the asymptotes of the hyperbola. Latus rectum of hyperbola is the line segment perpendicular to the transverse axis that passes through any of its foci & whose endpoints lie on hyperbola. . pornvds