JEE PYQ: Parabola Question 5
Question 5 - 2021 (25 Feb Shift 2)
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
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Answer: 9
Solution
Tangent to parabola: $y = mx + \frac{1}{m}$. This is also tangent to circle $(x-3)^2 + y^2 = 9$. Distance from $(3,0)$ to line $= 3$: $\frac{|3m + \frac{1}{m}|}{\sqrt{1+m^2}} = 3$. Solving: $m = \pm\frac{1}{\sqrt{3}}$. Taking $m = \frac{1}{\sqrt{3}}$ (first quadrant): tangent point on parabola $\left(\frac{1}{m^2}, \frac{2}{m}\right) = (3, 2\sqrt{3})$ and on circle $\left(\frac{3}{2}, \frac{3\sqrt{3}}{2}\right)$. So $2(a+c) = 2\left(\frac{3}{2} + 3\right) = 9$.
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