Suppose that the foci of the ellipse x ^ 2 9 + y ^ 2 5 = 1 are ( f _ 1 , 0)and ( f _ 2 , 0 ) where f _ 1 > 0and 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

Correct answer: 0 f _ 1 ≡ ( 2, 0 ) f _ 2 ≡ ( - 2, 0 ) Parabolas are y ^ 2 = 8x y ^ 2 = - 16x T _ 1 : y = mx + (2)/(m) 0 = - 4m + (2)/(m) ⇒ m ^ 2 = (1)/(2) T _ 2 : y = mx - (4)/(m) 2m - (4)/(m) = 0 ⇒ m ^ 2 = 2 Now, m _ 2 ^ 2 + 1 m _ 1 ^ 2 = 2 + 2 = 4

JEE Advanced 2015Paper 2

Exam: JEE Advanced
Year: 2015
Paper: Paper 2
Subject: Mathematics
Chapter: Parabola

JEE Advanced 2015 · Paper 2 · Mathematics · Parabola

Suppose that the foci of the ellipse x ^ 2 9 + y ^ 2 5 = 1 are ( f...

Suppose that the foci of the ellipse

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

are \left({f}_{1}, 0\right)and ( {f}_{2}, 0 )

where

{f}_{1} > 0and{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

( {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 Suppose that the foci of the ellipse x ^ 2 9 + y ^ 2 5 = 1 are ( f _ 1 , 0)and ( f _ 2 , 0 ) where f _ 1 > 0and f _ 2 <

Correct answer

Step-by-step solution

{f}_{1} ≡ ( 2, 0 )

{f}_{2} ≡ ( - 2, 0 )

Parabolas are

{y}^{2} = 8x

{y}^{2} = - 16x

{T}_{1}: y = mx + \dfrac{2}{m}

0 = - 4m + \dfrac{2}{m}

⇒ {m}^{2} = \dfrac{1}{2}

{T}_{2}: y = mx - \dfrac{4}{m}

2m - \dfrac{4}{m} = 0

⇒ {m}^{2} = 2

Now,

{m}_{2}^{2} + \dfrac{1}{{m}_{1}^{2}} = 2 + 2 = 4

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