Q. Consider the two curves $C_{1}:y^{2} = 4x$ $C_{2}:x^{2} + y^{2} - \ 6x + 1 = 0,$ then
A) $C_{1}$ and $C_{2}\$ touch each other only at one point
B) $C_{1}$ and $C_{2}\$ touch each other exactly at two points
C) $C_{1}$ and $C_{2}$ intersect (but do not touch) at exactly two point
D) $C_{1}$ and $C_{2}$ neither intersect nor touch each other
Explanation
(b)
For the points of intersection of the two given curves
$C_{1}:y^{2} = 4x$ and $C_{2}:x^{2} + y^{2} - 6x + 1 = 0$
We have, $x^{2} + 4x - 6x + 1 = 0\ \Rightarrow (x - 1)^{2} = 0$
$\Rightarrow \ \ x = 1,\ 1 \Rightarrow \ \ y = 2,\ - 2\$
Thus, the given curves touch each other at exactly two points $(1,\ 2)$ and$\ (a,\ - 2)$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.