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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=ax2+bx+cy = ax^2 + bx + c crosses the xx-axis at (α,0)(\alpha, 0) and (β,0)(\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:
Abca\sqrt{\dfrac{bc}{a}}
Bac2ac^2
Cba\dfrac{b}{a}
Dca\sqrt{\dfrac{c}{a}}correct
Solution
Step 1: Here α,β\alpha, \beta are the roots of ax2+bx+c=0ax^2 + bx + c = 0, so αβ=ca\alpha\beta = \dfrac{c}{a}. A circle through (α,0),(β,0)(\alpha, 0),(\beta, 0) is
S≡(x−α)(x−β)+y2+λy=0.S \equiv (x - \alpha)(x - \beta) + y^2 + \lambda y = 0.
 Step 2: The tangent length from the origin is S(0,0)=αβ=ca.\sqrt{S(0,0)} = \sqrt{\alpha\beta} = \sqrt{\dfrac{c}{a}}. Correct answer: (4)
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