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Two Circles Touching Internally and Externally

If a circle i...

Question

If a circle intersects the parabola $y^{2} = 4 a x$ at point $A (a t_{1}^{2}, 2 a t_{1}), B (a t_{2}^{2}, 2 a t_{2}), C (a t_{3}^{2}, 2 a t_{3}), D (a t_{4}^{2}, 2 a t_{4})$ , then $t_{1} + t_{2} + t_{3} + t_{4} + t_{4}$ is

A

1

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B

- 1

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C

0

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D

2

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Solution

The correct option is C $0$
Given, Parabola $y^{2} = 4 a x$

Let the equation of circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$

Since $(a t^{2}, 2 a t)$ lies on parabola then, (say general pont)

$⟹ (a t^{2})^{2} + (2 a t)^{2} + 2 g (a t^{2}) + 2 f (2 a t) + c = 0$

$⟹ a^{2} t^{4} + 4 a^{2} t^{2} + 2 a g t^{2} + 4 a f t + c = 0$

This equation is of fourth degree,
Therefore sum of roots os above equation is $0$ ., As the coefficient of $t^{3}$ is $0$ .

$∴ t_{1} + t_{2} + t_{3} + t_{4} = 0$

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Similar questions

Q. Assertion :If points

A (c t_{1}, \frac{c}{t_{1}})

B (c t_{2}, \frac{c}{t_{2}})

C (c t_{3}, \frac{c}{t_{3}})

D (c t_{4}, \frac{c}{t_{4}})

lies on the circle

x^{2} + y^{2} + 2 g x + 2 f y + k = 0

t_{1} + t_{2} + t_{3} + t_{4} = - \frac{2 g}{c}

. Reason: Points

A, B, C, D

are parametric points of

x y = c^{2}

.

Q. Match the column
The parabola

y^{2} = 4 a x

A B

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

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

Column - I	Column - II
(A) $A B$ is a normal chord if	(P) $t_{2} = - t_{1} - \frac{1}{2}$
(B) $A B$ is a focal chord	(Q) $t_{2} = \frac{- 4}{t_{2}}$
(C) $A B$ subtends $90^{\circ}$ at point $(0, 0)$ if	(R) $t_{2} = \frac{- 1}{t_{1}}$ (S) $t_{2} = - t_{1} - \frac{2}{t_{1}}$

A C

of the parabola

y^{2} = 4 a x

subtends angles of

90^{\circ}

B

D

on the parabola. If

A, B, C

D

are respresented by the points

t_{1}

t_{2}

t_{3}

t_{4}

respectively, then

t_{1}, t_{2}, t_{3}

be three points on the parabola

y^{2} = 4 x

. If the normal at

t_{1}

intersects the parabola at

t_{2}

and the normal at

t_{2}

intersects the parabola at

t_{3}

3 t_{1} + 13 t_{2} + 9 t_{3} = 0

Q. If four point

(t_{i}, \frac{1}{t_{i}}) (i = 1, 2, 3, 4)

are concyclic then

t_{1} t_{2} t_{3} t_{4} =

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