Question

Conisder the two curves : $c_1:y^2=4x:c_2:x^2+y^2-6x+1=0$

• 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 $C_2$ intersects (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 exactly at two points

$=>C_1:y^2=4x$ $(a=1).............\left(1\right)$

$=>C_2:x^2+Y^2-6x+1+0............\left(2\right)$

Point of interraction$=>x^2+4x-6x+1=0$

$=>x^2-2x+1=0$

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

$=>x=1$

$=>y=\pm 2$

Points$(1,2)$ and $(1,-2)$

Tangent to $\left(1\right)$ at $(1,2)=>2y=2(x+1)$

$=>y=x+1..........\left(3\right)$

Tangent $\left(2\right)$ at $(1,2)=>x+2y-3(x+1)+1=0$

$=>y=x+1.................\left(4\right)$

Line $3$ and Line $4$ are coincident$=>C_1$ and $C_2$ touch at $(1,2)$

Tangent to $\left(1\right)$ at $(1,-2)=>-2y=2(x+1)$

$=>x+y+1=0..............\left(5\right)$

Tangent to $\left(2\right)$ at $(1,-2)=>-2y+x-3(x+1)+1=0$

$=>-2y-2x-2=0$

$=>x+y+1=0...........\left(6\right)$

Line $5$ and Line $6$ are coincident$=>C_1$ and $C_2$ touch at $(1,-2)$

So, $C_1$ and $C_2$ touch each other at two points.