Question

The radius of circle, touching the parabola $y^{2}=8x$ $at$ $\left(2,4\right)$ and passing through $\left(0,4\right),$ is

• A. $1$ unit
• B. $2$ units
• C. $\sqrt{2}$ units
• D. $\sqrt{3}$ units

Answer 1

Answer: (C)

Equation of the tangent at $\left(2,4\right)$ on the parabola $y^{2}=8x$ is $y\left(4\right)=8\left(\frac{x + 2}{2}\right)\Rightarrow y=x+2$
Let equation of circle touching line $y=x+2$ at $\left(2,4\right)$ is $\left(x - 2\right)^{2}+\left(y - 4\right)^{2}+\lambda \left(x - y + 2\right)=0$ which passes through $\left(0,4\right)$ $\Rightarrow 4+0+\lambda \left(0 - 4 + 2\right)\Rightarrow \lambda =2$
$\Rightarrow $ Equation of the required circle is $x^{2}+y^{2}-2x-10y+24=0$
$\Rightarrow $ The radius of the circle is $\sqrt{1^{2} + 5^{2} - 24}=\sqrt{2}$