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Byju's Answer

Standard XII

Mathematics

Parametric Equation of Tangent

The tangents ...

Question

The tangents at the points $(a t_{1}^{2}, 2 a t_{1})$ , $(a t_{2}^{2}), 2 a t_{2})$ on the parabola $y^{2} = 4 a x$ are at right angles if

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Solution

The correct option is A
Tangent at $t_{1}$ is $y t_{1} = x + a t_{1}^{2}$
Tangent at $t_{2}$ is $y t_{2} = x + a t_{2}^{2}$
Tangents are at right angles $\Rightarrow t_{1} t_{2} = - 1$

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Similar questions

Q. If the tangents are drawn at

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

and

(a t_{2}^{2}, 2 a t_{2})

on the parabola

y^{2} = 4 a x

intersect on axis of the parabola, then

If the chord joining the points $(a t_{1}^{2}, 2 a t_{1})$ and $(a t_{2}^{2}, 2 a t_{2})$ of the parabola $y^{2} = 4 a x$ passes through the focus of the parabola, then

Q. Prove that the equation to the circle passing through the points

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

and

(a t_{2}^{2}, 2 a t_{2})

and the intersection of the tangents to the parabola at these points is

x^{2} + y^{2} - a x [(t_{1} + t_{2})^{2} + 2] - a y (t_{1} + t_{2}) (1 - t_{1} t_{2}) + a^{2} t_{1} t_{2} (2 - t_{1} t_{2}) = 0 .

Find the relation between $t_{1} a n d t_{2}$ if normals at $(a t_{1}^{2}, 2 a t_{1}) a n d (a t_{2}^{2}, 2 a t_{2})$ meet on the parabola.

Q. Let chord

A B

of the parabola

y^{2} = 4 x

subtend

90^{\circ}

at the vertex. If the coordinates of

A

and

B

are

(t_{1}^{2}, 2 t_{1})

and

(t_{2}^{2}, 2 t_{2})

respectively, then

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The tangents at the points (at21,2at1),(at22),2at2) on the parabola y2=4ax are at right angles if

The correct option is A Tangent at t1 is yt1=x+at21 Tangent at t2 is yt2=x+at22 Tangents are at right angles ⇒t1t2=−1

The tangents at the points $(a t_{1}^{2}, 2 a t_{1})$ , $(a t_{2}^{2}), 2 a t_{2})$ on the parabola $y^{2} = 4 a x$ are at right angles if

The correct option is A
Tangent at $t_{1}$ is $y t_{1} = x + a t_{1}^{2}$
Tangent at $t_{2}$ is $y t_{2} = x + a t_{2}^{2}$
Tangents are at right angles $\Rightarrow t_{1} t_{2} = - 1$