Question

If a circle $S=0$ touches the parabola $y^2=4x$ at the point $(1,-2)$ and is passing through the origin, then the radius of $S=0$ is equal to

• A. $5\sqrt{2}$
• B. $\frac{5}{\sqrt{2}}$
• C. $2\sqrt{5}$
• D. $\frac{2}{\sqrt{5}}$

Answer 1

The correct option is B $\frac{5}{\sqrt{2}}$

Equation of tangent to $y^2=4x$ at $(1,-2)$ is
$y(-2)=2(x+1)\Rightarrow x+y+1=0$

Equation of the circle is
$(x-1{)}^2+(y+2{)}^2+\lambda (x+y+1)=0$
It is passing through the origin, so
$1+4+\lambda =0\Rightarrow \lambda =-5$

Hence, the equation of the circle is
$(x-1{)}^2+(y+2{)}^2-5(x+y+1)=0\Rightarrow x^2+y^2-7x-y=0$

Hence, the radius of the circle
$=\sqrt{{(-\frac{7}{2})}^2+{(-\frac{1}{2})}^2}=\sqrt{\frac{50}{4}}=\frac{5}{\sqrt{2}}$