Pls answer this: Let tangents drawn from point (0,5) to the ellipse = 1 are perpendicular and meet major - Maths - Conic Sections

Question

Pls answer this:
Let tangents drawn from point (0,5) to the ellipse 
 = 1 are perpendicular and meet major axis of ellipse at points A
& B A hyperbola 'H' is drawn whose eccentricity is reciprocal
of eccentricity of ellipse & whose foci are points A & B,
then- (1) eccentricity of ellipse is 
 (2) hyperbola 'H' is 
 = 25 (3) hyperbola 'H' is 4x2 – 12y2 =
75 (4) Length of latus rectum of ellipse is

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

For x2a2+y2b2=1, perpendicular tangents are drawn from director circlewhose equation isx2+y2=a2+b2Here b2=5⇒x2+y2=a2+50,5 must lie on director circle⇒0+25=a2+5⇒a2=25-5⇒a2=20Hence ellipse is x220+y25=1e=a2-b2a2=20-520=1550=34⇒e=32Length of latus ractum=2b2a=2×520=1025=5

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