An ellipse with eccentricity 1 2 passes through P(3,4) whose nearer focus is S(0,0) and equation of tangent at P on ellipse is 3 x+4 y-25=0 . If a chord through S parallel to tangent at P intersects the ellipse at A and B then:

Step-by-step solution

P(3,4) is foot of perpendicular from S(0,0) on 3 x+4 y-25=0 which is also on ellipse P is vertex of ellipse. Distance between focus and directrix =a-a e=5 aligned & a- a 2 =5 a=10 & b^2=a^2 (1-e^2 )=100 (1- 1 4 )=75 aligned Clearly A B will be latus rectum of ellipse. A B= length of L R= 2 b^2 a =15 Focal length =2 a e=10 Mid-point of vertex and centre will focus. centre =(-3,-4)