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 also passes through these two points. The length of a tangent from the origin to the circle is ?

Answer 1

Answer:

Step-by-step explanation:

Here draw a circle with the radius of

Answer 2

Answer:

Step-by-step explanation:

The equation of the parabola $y=ax^{2}+bx+c$

The two value of $x =\alpha$ and $x =\beta$

For Quadratic Equation $ax^{2}+bx+c=0 \text{ [where x is the variable and a, b and c are known values]}$

The value of $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

$b^{2}-4ac=\text{ is called Discriminant (D)}$

Therefore the two value of  $\alpha=\frac{-b+\sqrt{D}}{2a}$ and $\beta=\frac{-b-\sqrt{D}}{2a}$

therefore,

$\alpha\times\beta=\frac{-b+\sqrt{D}}{2a}\times\frac{-b-\sqrt{D}}{2a}\\\\\Rightarrow\alpha\times\beta=-\frac{1}{4a^{2}}(\sqrt{D}-b)(\sqrt{D}+b)\\\\\Rightarrow\alpha\times\beta=-\frac{1}{4a^{2}}[(\sqrt{D})^{2}-b^{2}]\\\\\Rightarrow\alpha\times\beta=-\frac{1}{4a^{2}}[(D-b^{2}]\\\\\Rightarrow\alpha\times\beta=-\frac{1}{4a^{2}}(b^{2}-4ac-b^{2}]=\frac{c}{a}$

The length of a tangent from the origin = OT

From Secant-Tangent property of circle, If OT is the tangent and OAB is the secant (line segment intersects the circle at two points) to the circle intersecting the circle at points A and B , then $OT^{2}=OA\times{OB}$ where OT is the length of the tangent from O to the circle.

$OT^{2}=OA\times{OB}\\\\\Rightarrow{OT}=\sqrt{OA\times{OB}}\\\\\Rightarrow{OT}=\sqrt{\alpha\times\beta}=\sqrt{\frac{c}{a}}$