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Byju's Answer

Standard XIII

Mathematics

Focii of Ellipse

Suppose that ...

Question

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) 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 $(\frac{1}{m_{1}^{2}} + m_{2}^{2})$ is

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Solution

The correct option is A 4

Tangent to $P_{1}$ passes through $(2 f_{2}, 0) i . e . (- 4, 0) .$ .
$∴ T_{1} : y = m_{1} x + \frac{2}{m_{1}} \Rightarrow 0 = - 4 m_{1} + \frac{2}{m_{1}} \Rightarrow m_{1}^{2} = \frac{1}{2}$
Also tangent to $P_{2}$ passes through $(f_{1}, 0) i . e . (2, 0) .$
$\Rightarrow t_{2} : y = m_{2} x + \frac{- 4}{m_{2}} \Rightarrow 0 = 2 m_{2} - \frac{4}{m_{2}} \Rightarrow m_{2}^{2} = 2 ∴ \frac{1}{m_{1}^{2}} + m_{2}^{2} = 2 + 2 = 4$

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Similar questions

Q. 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})

is .

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 $(\frac{1}{m_{1}^{2}} + m_{2}^{2})$ is

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The correct option is A 4 Tangent to P1 passes through (2f2,0)i.e.(−4,0).. ∴T1:y=m1x+2m1⇒0=−4m1+2m1⇒m21=12 Also tangent to P2 passes through (f1,0)i.e.(2,0). ⇒t2:y=m2x+−4m2⇒0=2m2−4m2⇒m22=2∴1m21+m22=2+2=4