Question 1

The eccentricity of the ellipse whose major axis is three times the minor axis is:

Accepted Answer

Let a be the major axis and b, the minor axis of the ellipse, then 3 minor axis = major axis. 3 b=a Eccentricity is given by: array l b^ 2 =a^ 2 (1-e^ 2 ) \ b^ 2 =9 b^ 2 (1-e^ 2 ) \ 1 9 = (1-e^ 2 ) \ e^ 2 = 8 9 \ e= 2 2 3 array

Question 2

An ellipse has OB as semi minor axis, F and F ^ its focii and the angle FBF' is a right angle. Then the eccentricity of the ellipse is

Accepted Answer

FBF ^ =90^ FB ^ 2 + F ^ B ^ 2 = FF ^ 2 ( a^ 2 e^ 2 +b^ 2 )^ 2 + ( a^ 2 e^ 2 +b^ 2 )^ 2 =(2 a e)^ 2 2 (a^ 2 e^ 2 +b^ 2 )=4 a^ 2 e^ 2 e^ 2 = b^ 2 a^ 2 Also e^ 2 =1- b^ 2 a^ 2 =1-e^ 2 array l 2 e^ 2 =1 \ e= 1 2 array

Question 3

If x ma + y nb =1 touches the ellipse x^ 2 a^ 2 + y^ 2 b^ 2 =1, then

Accepted Answer

The line is y=- nb ma x-n b. It will touch the ellipse if (-n b)^ 2 =a^ 2 (- n b m a )^ 2 +b^ 2 [ c ^ 2 = a ^ 2 ~m ^ 2 + b ^ 2 ] n^ 2 = n^ 2 m^ 2 +1 m ^ 2 = n ^ 2 n ^ 2 -1 or n ^ 2 = m ^ 2 ~m ^ 2 -1

Ellipse — Previous Year Questions & Solutions