Question

The number of points at which the parabola y² = 4x and the circle x² + y² - 6x + 1 = 0 touch each other is ...... 

Answer 1

Substituting y² = 4x in the given circle equation, we have x² - 2x  + 1 = 0 and hence (1, ± 2) are common points of the two curves. Also at P(1, 2), equation of the tangent to the parabola is

y(2) - 2(x + 1) = 0

 ⇒ y - x - 1 = 0   ..(i)

The centre and the radius of the circle are (3, 0) and 2√2, respectively. Now, the distance of the centre (3, 0) from Eq. (1) is

Hence, the line provided in Eq. (1) also touches the circle in a similar manner as the two curves touch at Q.