Parabola
Common tangent to circle and parabola, chord length
nta_pyq_2023_jan
Grade 11
Question:
Let A be a point on the x-axis. Common tangents are drawn from A to the curves $x^2 + y^2 = 8$ and $y^2 = 16x$. If one of these tangents touches the two curves at Q and R, then $(QR)^2$ is equal to
Step-by-Step Solution
Key Concept: Find tangent from x-axis point A to both curves simultaneously; use tangent to parabola $y = mx + 4/m$ and tangency condition to circle
Tangent to $y^2=16x$: $y = mx+4/m$. Tangency to $x^2+y^2=8$: $|4/m|/\sqrt{1+m^2} = 2\sqrt{2} \Rightarrow m = \pm 1$. For $m=1$: tangent $y=x+4$, touches parabola at $R(4,8)$ (using $a/m^2 = 4, 2a/m = 8$, $a=4$). Touches circle at $Q(-2,2)$. $(QR)^2 = (4+2)^2+(8-2)^2 = 36+36=72$. Answer: (4)
Correct Answer: 72