Prove that the curves y 2 = 4x and x 2 + y 2 – 6x + 1 = 0 touch each other at the point (1, 2).

Dear Student

It is given that
curve equations are:y^2= 4x....(1)
and
x^2+y^2–6x+ 1 = 0..... (2)
differentiating (i) w.r.t. x, we get

2y.(dy/dx) = 4
⇒dy/dx = 2/y
Slope of tangent at (1, 2),
m1= 2/2 = 1
Differentiating (ii) w.r.t. x, we get

2x + 2y.(dy/dx)–6 = 0
2y. dy/dx = 6–2x
⇒dy/dx = (6–2x)/ 2y
Hence, the slope of the tangent at the same point (1, 2)
⇒m2= (6–2 x 1)/ (2 x 2) = 4/4 = 1
It can be seen that
m1= m2= 1 at the point (1, 2).
Thus, the given circles touch each other at the same point (1, 2).


Thanks