Consider two curves `C1:y^2=4x` ; `C2=x^2+y^2-6x+1=0`. Then, a. C1 and C2 touch each other at one point b. C1 and C2 touch each other exactly at two point c. C1 and C2 intersect(but do not touch) at exactly two point d. C1 and C2 neither intersect nor touch each other

Options

A. `C_(1)" and "C_(2)` touch each other at one point
B. `C_(1)" and "C_(2)` touch eacth other exactly at two point
C. `C_(1)" and "C_(2)` intersect ( but do not touch) at exactly two points
D. `C_(1)" and "C_(2)` neither intersect not touch each other

Answer

B

Explanation

The x-coordinates of the points of intersection of `C_(1)" and "C_(2)` are roots of the equation
`x^(2)+4x-6x+2=0rArrx=1`
Putting x = 1 in `y^(2)=4x`, we get `y = +- 2`.
Thus, two curves intersect at (1, 2) and (1, -2).
Equations of tangents to `C_(1)" and "C_(2)` at (1, 2) are
`2y=2(x+1)" and "x+2y-3(x+1)+1=0`
`"or, y=x+1 and y=x+1"`
Thus, `C_(1)" and "C_(2)` touch each other at P(1, 2). Similarly, `C_(1)" and C_(2)` have the same tangent x + y + 1 = 0 at Q(1, -2).