Question

A parabola y=ax2+bx+cy=ax2+bx+c crosses the x−axisx−axis at (α,0)(α,0) and (β,0)(β,0) both to the right of the origin. A circle also passes through these two points. The length of a tangent from the origin to the circle is

Answer 1

Answer:

√αβ

Step-by-step explanation:

Given that the prqabola intersects the x-axis at points A(α,0) and B(β,0).

Also given that a circle passes through both the points A and B , hence OAB (x-axis is the secant to that circle).

We need to find the length of the tangent from origin to circle,

From Secant-Tangent property of circle, If PT is the tangent and PAB is the secant to the circle intersecting the circle at points A and B , then PT²=PA*PB where PT is the length of the tangent from P to the circle.

Here P is origin (0,0), A is (α,0) and B is (β,0).

Also, both α and β are positive since they lie to right side of the origin.

so, PA = α and PB=β

Now,using Secant-tangent theorem, we get PT²=αβ.

Thus, PT = √αβ.