Equation of Hyperbola

1. Analytical Geometry T- 1-855-694-8886 Email- [email protected] T- 1-855-694-8886 Email- [email protected] By iTutor.com

2. Hyperbola A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. A hyperbola is a curve where the distances of any point from: a fixed point (the  focus ), and a fixed straight line (the  directrix ) are always in the same ratio. Hyperbola focus Directrix These distance are always in same ratio. focus Directrix © iTutor . 2000-2013. All Rights Reserved

3. The two fixed points are called the foci of the hyperbola. The mid-point of the line segment joining the foci is called the centre of the hyperbola . The line through the foci is called the transverse axis and the line through the centre and perpendicular to the transverse axis is called the conjugate axis . The points at which the hyperbola intersects the transverse axis are called the vertices of the hyperbola. Hyperbola focus Directrix focus Directrix Hyperbola vertex vertex conjugate axis T ransverse axis Centre The " asymptotes“ are not part of the hyperbola, but show where the curve would go if continued indefinitely in each of the four directions. Asymptotes Asymptotes © iTutor . 2000-2013. All Rights Reserved

4. We denote the distance between the two foci by 2c , the distance between two vertices by 2a and we define the quantity b as Also 2b is the length of the conjugate axis To find the constant PF – PG: By taking the point P at A and B in the, we have BF – BG = AG – AF (by the definition of the hyperbola) BA + AF – BG = AB + BG – AF i.e., AF = BG , so that , BF – BG = BA + AF – BG = BA = 2a 2c 2a a X X’ Y Y’ F G A B b c

5. Standard equation of Hyperbola The equation of a hyperbola is simplest if the centre of the hyperbola is at the origin and the foci are on the x-axis or y-axis. We will derive the equation for the hyperbola shown in with foci on the x-axis. Let F and G be the foci and O be the mid-point of the line segment FG. Let O be the origin and the line through O through G be the positive x-axis and that through F , as the negative x axis. The line through O  to the x-axis be the y-axis. G (c , 0) F (-c , 0 ) P (x , y) x = -a x = a O X X’ Y Y’ © iTutor . 2000-2013. All Rights Reserved

6. Let the coordinates of F be (–c, 0) and G be (–c, 0) . Let P(x , y) be any point on the hyperbola such that the difference of the distances from P to the farther point minus the closer point be 2a. So given, PF – PG = 2a Using the distance formula, we have Squaring both side we get square binomials or or © iTutor . 2000-2013. All Rights Reserved

7. By Squaring i.e. , Hence any point on the hyperbola satisfies or or or (Since b 2 = c 2 – a 2 ) © iTutor . 2000-2013. All Rights Reserved

8. Conversely, let P( x, y) satisfy the above equation with 0 < a< c. Then, Therefore, Similarly, (Since b 2 = c 2 – a 2 ) © iTutor . 2000-2013. All Rights Reserved

9. In hyperbola c> a; and since P is to the right of the line x= a, x> a, Therefore, becomes negative. Thus, Therefore, Also, note that if P is to the left of the line x = –a, then In that case PF – PG = 2 a So, any point that satisfies lies on hyperbola Thus, we proved that the equation of hyperbola with origin (0,0) and transverse axis along x-axis is © iTutor . 2000-2013. All Rights Reserved

10. The equation for a hyperbola can be derived by using the definition and the distance formula. The resulting equation is: a a c b b This looks similar to the ellipse equation but notice the sign difference. To graph a hyperbola, make a rectangle that measures 2a by 2b as a sketching aid and draw the diagonals. These are the asymptotes. © iTutor . 2000-2013. All Rights Reserved

11. Standard equation of a hyperbola with its center at the origin and vertical transverse axis For a hyperbola with its center at the origin and has the transverse axis horizontal, the standard equation is: c 2 = a 2 + b 2 The equations of its asymptotes are: Focus: (0, c) (0, - c) (0, c) vertices: (0, a) (0, - a) (0, a) (0, - c) (0, c) (0, - a) (0, a) © iTutor . 2000-2013. All Rights Reserved

12. The center of the hyperbola may be transformed from the origin. The equation would then be: horizontal transverse axis vertical transverse axis The axis is determined by the first term NOT by which denominator is the largest. If the x term is positive it will be horizontal , if the y term is the positive term it will be vertical. © iTutor . 2000-2013. All Rights Reserved

13. Eccentricity   The  eccentricity  (usually shown as the letter e), it shows how " uncurvy " (varying from being a circle) the hyperbola is. On this diagram: P is a point on the curve, F is the focus and N is the point on the directrix so that PN is perpendicular to the directrix. The ratio PF/PN is the   eccentricity  of the hyperbola (for a hyperbola the eccentricity is always greater than 1). It can also given by the formula: Hyperbola F P N Directrix © iTutor . 2000-2013. All Rights Reserved

14. Latus rectum of Hyperbola Latus rectum of hyperbola is a line segment perpendicular to the transverse axis through any of the foci and whose end points lie on the hyperbola. Directrix F Latus rectum The length of the Latus Rectum is © iTutor . 2000-2013. All Rights Reserved

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