Suppose that the foci of the ellipse \(\frac{x^{2}}{9}+\frac{y^{2}}{5}=1\) are \(\left(f_{1}, 0\right)\) and \(\left(f_{2}, 0\right)\) 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 \(\left(f_{1}, 0\right)\) and \(\left(2 f_{2}, 0\right)\), respectively. Let \(T_{1}\) be a tangent to \(P_{1}\) which passes through \(\left(2 f_{2}, 0\right)\) and \(T_{2}\) be a tangent to \(P_{2}\) which passes through \(\left(f_{1}, 0\right)\). If \(m_{1}\) is the slope of \(T_{1}\) and \(m_{2}\) is the slope of \(T_{2}\), then the value of \(\left(\frac{1}{m_{1}^{2}}+m_{2}^{2}\right)\) is

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Parabola

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

(2 f_{2}, 0)

, respectively. Let

T_{1}

be a tangent to

P_{1}

which passes through

(2 f_{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

(\frac{1}{m _{1}^{2}} + m_{2}^{2})

Answer: 4

For the ellipse

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

, we have

a^{2} = 9

b^{2} = 5

, so

e^{2} = 1 - \frac{5}{9} = \frac{4}{9}

, giving

e = \frac{2}{3}

. Thus

f_{1} = 2

f_{2} = - 2

. Parabola

P_{1}

has focus

(2, 0)

, so

y^{2} = 8 x

. Tangent

y = m_{1} x + \frac{2}{m _{1}}

passes through

(- 4, 0)

0 = - 4 m_{1} + \frac{2}{m _{1}}

, giving

m_{1}^{2} = \frac{1}{2}

. Parabola

P_{2}

has focus

(- 4, 0)

, so

y^{2} = - 16 x

. Tangent

y = m_{2} x - \frac{4}{m _{2}}

passes through

(2, 0)

0 = 2 m_{2} - \frac{4}{m _{2}}

, giving

m_{2}^{2} = 2

. Therefore,

\frac{1}{m _{1}^{2}} + m_{2}^{2} = 2 + 2 = 4

JEE Advanced · 2015 · 24 May 2015 · Paper 2 · Integer answer