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Standard XII

Mathematics

Average Rate of Change

If st =1, t...

Question

If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

\frac{(t^{2} + 1)^{2}}{2 t^{3}}

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\frac{a (t^{2} + 1)^{2}}{2 t^{3}}

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\frac{a (t^{2} + 1)^{2}}{t^{3}}

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\frac{a (t^{2} + 2)^{2}}{t^{3}}

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Solution

The correct option is C $\frac{a (t^{2} + 1)^{2}}{2 t^{3}}$
Tangent at $P : t y = x + a t^{2}$ or $y = \frac{x}{t} + a t$
Normal at $S : y + \frac{x}{t} = \frac{2 a}{t} + \frac{a}{t^{3}}$
Solving, $2 y = a t + \frac{2 a}{t} + \frac{a}{t^{3}}$
$y = \frac{a (t^{2} + 1)^{2}}{2 t^{3}}$

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Similar questions

Q. Let

a, s, t

be nonzero real numbers. Let

P (a t^{2}, 2 a t), and S (a s^{2}, 2 a s)

be distinct points on the parabola

y^{2} = 4 a x

. If

s t = 1

, then the tangent at

P

and the normal at

S

to the parabola meet at a point whose ordinate is

The area of the triangle formed by three points $P {a t_{1} t_{2}, a (t_{1} + t_{2})}, Q {a t_{2} t_{3}, a (t_{2} + t_{3})}, R {a t_{3} t_{1}, a (t_{3} + t_{1})}$ is _____.

Q. The vertices of a triangle are

[a t_{1} t_{2}, a (t_{1} + t_{2})], [a t_{2} t_{3}, a (t_{2} + t_{3})], [a t_{3} t_{1}, a (t_{3} + t_{1})]

. Find the coordinates of its orthocentre.

Q. If

t_{1} + t_{2} + t_{3} = - t_{1} t_{2} t_{3}

, then the orthocentre of the triangle formed by the points

(a t_{1} t_{2}, a (t_{1} + t_{2}))

(a t_{2} t_{3}, a (t_{2} + t_{3}))

and

(a t_{3} t_{1}, a (t_{3} + t_{1}))

lies on:

Q. The point of concurrency of the altitudes drawn from the vertices

A (a t_{1} t_{2}, a (t_{1} + t_{2})), B (a t_{2} t_{3}, a (t_{2} + t_{3}))

and

C (a t_{3} t_{1}, a (t_{3} + t_{1}))

of the triangle

A B C

(where

t_{1} \neq t_{2} \neq t_{3})

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If st=1, then the tangent at P and the normal at S to the parabola meet at a point whose ordinate is

The correct option is C a(t2+1)22t3Tangent at P:ty=x+at2 or y=xt+atNormal at S:y+xt=2at+at3Solving, 2y=at+2at+at3y=a(t2+1)22t3

If $s t = 1$ , then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

The correct option is C $\frac{a (t^{2} + 1)^{2}}{2 t^{3}}$
Tangent at $P : t y = x + a t^{2}$ or $y = \frac{x}{t} + a t$
Normal at $S : y + \frac{x}{t} = \frac{2 a}{t} + \frac{a}{t^{3}}$
Solving, $2 y = a t + \frac{2 a}{t} + \frac{a}{t^{3}}$
$y = \frac{a (t^{2} + 1)^{2}}{2 t^{3}}$