Question

If the two curves are given as $C_1:y^2=4x,C_2:x^2+y^2-6x+1=0$, 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 points.
• D. $C_1$ and $C_2$ neither intersect nor touch each other.

Answer 1

Answer: (B)

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

$C_1:y^2=4x$

$C_2:x^2+y^2-6x+1=0$

Substituting $y^2=4x$ in $C_2$, we get

$x^2+4x-6x+1=0$

$\Rightarrow (x-1{)}^2=0$

$\Rightarrow x=1$

At $x=1;y=\pm 2$

The circle and the parabola touch each other at $x=1$ i.e. at the points $(1,2)$ and $(1,-2)$ as shown in the figure.

Hence, option B.