JEE Advance - Mathematics (2008 - Paper 1 Offline - No. 11)

Consider the two curves $${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then,

$${C_1}$$ and $${C_2}$$ touch each other only at one point.

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

$${C_1}$$ and $${C_2}$$ intersect (but do not touch ) at exactly two points

$${C_1}$$ and $${C_2}$$ neither intersect nor touch each other

Given that,

$${C_1}:{y^2} = 4x$$

$${C^2}:{x^2} + {y^2} - 6x + 1 = 0$$

Putting $${y^2} = 4x$$ in $${x^2} + {y^2} - 6x + 1 = 0$$, we get

$${x^2} + 4x - 6x + 1 = 0 \Rightarrow {(x - 1)^2} = 0$$

$$x = 1$$

putting $$x = 1$$ in $${y^2} = 4x$$ we get $$y = \, \pm ~2$$

IIT-JEE 2008 Paper 1 Offline Mathematics - Ellipse Question 9 English Explanation

So, the curves touches each other at two points (1, 2) and (1, $$-$$2),