Suppose that the foci of the ellipse are and where and

MATHEMATICS · Subjective · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Suppose that the foci of the ellipse

\frac{x^2}{9} + \frac{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( \frac{1}{m_1^2} + m_2^2 \right)