Question

Consider the two curves$C_1:y^2=4xand{C}_2:x^2+y^2-6x+1=0$, then

• A. $C_1$and$C_2$ touch each other at only one point
• B. $C_1$and$C_2$ touch each other at exactly two points
• C. $C_1$and$C_2$ intersect (but do not touch ) at exactly two points
• D. $C_1$and$C_2$ neither intersect nor touch each other

Answer 1

The correct option is B

$C_1$and$C_2$ touch each other at exactly two points

Finding the intersection points of the given curves.

Consider the curve$C_2:x^2+y^2-6x+1=0$ and$C_1:y^2=4x$,

the solutions of these two curves give us the points of intersection.

$\begin{array}{rcl}\therefore x^2+4x-6x+1& =& 0\{y^2=4x\}\\ x^2-2x+1& =& 0\\ {(x-1)}^2& =& 0\\ x& =& 1,1\\ y& =& 2,-2\end{array}$

The given curves touch each other at exactly two points$(1,2)$ and$(1,-2)$.

Hence, the correct option is$\left(B\right)$.