Question

Consider the circle $C:x^2+y^2-6y+4=0$ and the parabola $P:y^2=x$ then

• A. The number of common tangents to $C$ and $P$ is $3$
• B. The number of common tangents to $C$ and $P$ is $2$
• C. $x-2y+1=0$ is one of the common tangents
• D. $x+2y+1=0$ is also one of the common tangents

Answer 1

Answer: (A), (C)

The correct options are
A The number of common tangents to $C$ and $P$ is $3$
C $x-2y+1=0$ is one of the common tangents

$C:x^2+(y-3{)}^2=5$
Centre of a circle is $(0,3)$
Let $Q(t^2,t)$ be any point on the parabola $y^2=x$ so that the equation of the tangent at $Q$ is $x-2ty+t^2=0$ which touches the circle $C.$
So, $\left|\frac{0-2t\left(3\right)+t^2}{\sqrt{1+4t^2}}\right|=\sqrt{5}$
$\Rightarrow (t^2-6t{)}^2=5(1+4t^2)$
$\Rightarrow t^4-12t^3+16t^2-5=0$
$\Rightarrow (t-1{)}^2(t^2-10t-5)=0$
$\Rightarrow t=1,5\pm \sqrt{30}$
Hence, the number of common tangent is $3$ and $x-2y+1=0$ is a common tangent when $t=1.$