Question

The number of integral points on the circle, touching the parabola $y^{2}=8x$ at $\left(2 , 4\right)$ and passing through $\left(0 , 4\right),$ are equal to

• A. $1$
• B. $2$
• C. $4$
• D. $3$

Answer 1

Answer: (C)

Equation of 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 the equation of the 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 $ Required circle is $x^{2}+y^{2}-2x-10y+24=0$
$\Rightarrow \left(x - 1\right)^{2}+\left(y - 5\right)^{2}=2$
If $x$ and $y$ are integers, then
$\left(x - 1\right)^{2}=1=\left(y - 5\right)^{2}$
$\Rightarrow x=0,2$ and $y=4,6$
$\Rightarrow 4$ integral points lie on the circle