Question

A line is a common tangent to the circle $(x-3)^2+y^2=9$ and the parabola $y^2=4x$. If the two points of contact (a, b) and (c, d) are distinct and lie in the first quadrant, then 2(a+c) is equal to ________ .

Hint:

The condition for a line $y=mx+c$ to be tangent to a standard parabola $y^2=4ax$ is $c=a/m$. For a circle $(x-h)^2+(y-k)^2=r^2$, the perpendicular distance from the center $(h,k)$ to the line must be equal to the radius $r$. These two conditions are key for solving common tangent problems.

Answer 1

Correct Answer: 9

Tangent to parabola: \[ y=mx+\frac{1}{m} \] Distance from center \((3,0)\) to line \(=3\): \[ \frac{|3m^2+1|}{\sqrt{m^4+m^2}}=3 \Rightarrow m=\frac1{\sqrt3} \] Point on parabola: \[ (c,d)=(3,2\sqrt3) \] Point on circle: \[ (a,b)=\left(\frac32,\frac{3\sqrt3}{2}\right) \] \[ 2(a+c)=9 \] \[ \boxed{9} \]