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Tangent Length from Origin to Circle Through Parabola Roots | JEE

JEE Maths question with a full step-by-step solution.

Question

A parabola

y = ax^2 + bx + c

crosses the

x

-axis at

(\alpha, 0)

and

(\beta, 0)

, both to the right of the origin. A circle passes through these two points. The length of the tangent from the origin to the circle is:

\sqrt{\dfrac{bc}{a}}

ac^2

\dfrac{b}{a}

\sqrt{\dfrac{c}{a}}

correct

Solution

Step 1: Here

\alpha, \beta

are the roots of

ax^2 + bx + c = 0

, so

\alpha\beta = \dfrac{c}{a}

. A circle through

(\alpha, 0),(\beta, 0)

S \equiv (x - \alpha)(x - \beta) + y^2 + \lambda y = 0.

Step 2: The tangent length from the origin is

\sqrt{S(0,0)} = \sqrt{\alpha\beta} = \sqrt{\dfrac{c}{a}}.

Correct answer: (4)

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