Question 4

x2 = - 16 y 5. y2 = 10x 6. x2 = - 9 y In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions: We denote the length of the major axis by 2a, the length of the minor axis by 2b and the distance between the foci by 2c. Thus, the length of the semi major axis is a and semi-minor axis is b (Fig10.22). 7. Focus (6,0); directrix x = - 6 8. Focus (0,-3); directrix y = 3 9. Vertex (0,0); focus (3,0) 10. Vertex (0,0); focus (-2,0) 11. Vertex (0,0) passing through (2,3) and axis is along x-axis. 12. Vertex (0,0), passing through (5,2) and symmetric with respect to y-axis. 10. 5 Ellipse Definition 4 An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is a constant. The two fixed points are called the foci (plural of ‘focus’) of the ellipse (Fig10.20). ANote The constant which is the sum of the distances of a point on the ellipse from the two fixed points is always greater than the distance between the two fixed points. The mid point of the line segment joining the foci is called the centre of the ellipse. The line segment through the foci of the ellipse is called the major axis and the line segment through the centre and perpendicular to the major axis is called the minor axis. The end points of the major axis are called the vertices of the ellipse(Fig 10.21). 10.5.1 Relationship between semi-major axis, semi-minor axis and the distance of the focus from the centre of the ellipse (Fig 10.23). Take a point P at one end of the major axis. Sum of the distances of the point P to the foci is F1 P + F2P = F1O + OP + F 2P (Since, F1P = F 1O + OP) = c + a + a - c = 2a Take a point Q at one end of the minor axis. Sum of the distances from the point Q to the foci is F1Q + F 2Q = 2 22 2 c b c b+ + + = 2 22 c b+ Since both P and Q lies on the ellipse. By the definition of ellipse, we have 2 2 2c b+ = 2a, i.e., a = 2 2c b+ or a 2 = b2 + c2 , i.e., c = 2 2b a− . 10.5.2 Eccentricity Definition 5 The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of the vertices of the ellipse (eccentricity is denoted by e) i.e., ce a= . Then since the focus is at a distance of c from the centre, in terms of the eccentricity the focus is at a distance of ae from the centre. 10.5.3 Standard equations of an ellipse The equation of an ellipse is simplest if the centre of the ellipse is at the origin and the foci are on the x-axis or y-axis. The two such possible orientations are shown in Fig 10.24. We will derive the equation for the ellipse shown above in Fig 10.24 (a) with foci on the x-axis. (a) Let F1 and F2 be the foci and O be the mid-point of the line segment F1F2. Let O be the origin and the line from O through F 2 be the positive x-axis and that through F1as the negative x-axis. Let, the line through O perpendicular to the x-axis be the y-axis. Let the coordinates of F1 be (- c, 0) and F2 be (c, 0) (Fig 10.25). Let P(x, y) be any point on the ellipse such that the sum of the distances from P to the two foci be 2a so given PF1 + PF 2 = 2a. ... (1) Using the distance formula, we have 2 22 2 ) () ( y c x y c x+ − + + + = 2a i.e., 2 2) ( y c x+ + = 2a - 2 2) ( y c x+ − Squaring both sides, we get (x + c)2 + y2 = 4a2 - 4 a 2 22 2 ) ( ) (y c x y c x+ − + + − 2 2 2 2 1x y a b + = which on simplification gives x a ca y c x− = + −2 2) ( Squaring again and simplifying, we get 2 2 2/2 c a y a x −+ = 1 i.e., 2 2/2 b y a x + = 1 (Since c 2 = a2 - b2) Hence any point on the ellipse satisfies 2/2 2/2 b y a x + = 1. ... (2) Conversely, let P (x, y) satisfy the equation (2) with 0 < c < a. Then y2 = b2       − 2 1 a x Therefore, PF1 = 2 2( )x c y+ + =       −+ + 2 2 2 2 2) ( a

= / 2 2 / 2 2 2 2( ) ( ) a xx c a c a   −+ + −    (since b2 = a2 - c2) =

a a   + = +   Similarly PF2 = ca x a− Hence PF1 + PF 2 = 2c ca x a - x aa a+ + = ... (3) So, any point that satisfies 2 2/2 b y a x + = 1, satisfies the geometric condition and so P(x, y) lies on the ellipse. Hence from (2) and (3), we proved that the equation of an ellipse with centre of the origin and major axis along the x-axis is 2 2 2 2 x y a b + = 1. Discussion From the equation of the ellipse obtained above, it follows that for every point P (x, y) on the ellipse, we have 2/2 2/2 b y a x − = ≤ 1, i.e., x2 ≤ a2, so - a ≤ x ≤ a. Therefore, the ellipse lies between the lines x = - a and x = a and touches these lines. Similarly, the ellipse lies between the lines y = - b and y = b and touches these lines. Similarly, we can derive the equation of the ellipse in Fig 10.24 (b) as 2 2 2 2 1x y b a + = . These two equations are known as standard equations of the ellipses. ANote The standard equations of ellipses have centre at the origin and the major and minor axis are coordinate axes. However, the study of the ellipses with centre at any other point, and any line through the centre as major and the minor axes passing through the centre and perpendicular to major axis are beyond the scope here. From the standard equations of the ellipses (Fig10.24), we have the following observations: