JEE Advanced 2015 Paper 2 Mathematics Q4

Mathematics

Problem

Suppose that the foci of the ellipse

\dfrac{x^2}{9}+\dfrac{y^2}{5}=1

are

(f_1,0)

and

(f_2,0)

, where

f_1>0

and

f_2<0

. Let

P_1

and

P_2

be two parabolas with a common vertex at

(0,0)

and with foci at

(f_1,0)

and

(2f_2,0)

, respectively. Let

T_1

be a tangent to

P_1

which passes through

(2f_2,0)

and

T_2

be a tangent to

P_2

which passes through

(f_1,0)

. If

m_1

is the slope of

T_1

and

m_2

is the slope of

T_2

, then the value of

\left(\dfrac{1}{m_1^2}+m_2^2\right)

is

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