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