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Standard XII

Mathematics

Parametric Equation of Parabola

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 $(0, 0)$ 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

$4$

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$2$

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$6$

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$8$

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Solution

The correct option is A
$4$

$\frac{x^{2}}{9} + \frac{y^{2}}{5} = 1, e = \sqrt{1 - \frac{5}{9}} = \frac{2}{3}$
Focus $(\pm a e, 0) = (\pm 2, 0)$
$f_{1} = 2, f_{2} = - 2$
Parabola $P_{1} = y^{2} = 4 \times 2 x$
$y^{2} = 8 x \dots (1)$
Parabola $P_{2} = y^{2} = - 4 \times 4 x$
$y^{2} = - 16 x \dots (2)$

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Similar questions

Suppose that the foci of the ellipse are (f₁, 0) and (f₂, 0) where f₁ >0 and f₂ <0 .Let P₁ and P₂ be two parabolas with a common vertex at (0, 0) and with foci at (f₁, 0) and (2f₂, 0), respectively. Let T₁ be a tangent to P₁ which passes through (2f₂, 0) and T₂ be a tangent to P₂ which passes through (f₁, 0). If m₁ is the slope of T₁ and m₂ is the slope of T₂, then the value of is

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 the 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})

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) 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})

A particle is observed from two frames $S_{1} a n d S_{2}$ . The frame $S_{2}$ moves with respect to $S_{1}$ with an acceleration a. Let $F_{1} a n d F_{2}$ be the pseudo forces on the particle when seen from $S_{1} a n d S_{2}$ respectively. Which of the following are not possible?

A particle is observed from two frames S1 and S2. The graph of relative velocity of S1 with respect to S2 is shown in figure. Let F1 and F2 be the pseudo forces on the particle when seen from S1 and S2 respectively. Which one of the following is not possible?

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The correct option is A 4 x29+y25=1, e=√1−59=23 Focus (±ae,0)=(±2,0) f1=2,f2=−2 Parabola P1=y2=4×2x y2=8x⋯(1) Parabola P2=y2=−4×4x y2=−16x⋯(2)