Question

Find the points on the curve x 2 + y 2 − 2 x − 3 = 0 at which the tangents are parallel to the x -axis.

Answer 1

Equation of the curve is given as,

x 2 + y 2 −2x−3=0

The slope of the tangent to the curve is given as,

slope =( dy dx )

Hence, the slope of the tangent of the given curve is given by differentiating with respect to x.

2x+2y( dy dx )−2=0 y( dy dx )=1−x dy dx = 1−x y

When the tangent is parallel to x axis, the slope of the tangent must be 0.

1−x y =0 x−1=0 x=1

Coordinate of y when x=1 is,

( 1 ) 2 + y 2 −2( 1 )−3=0 y 2 =4 y=±2

Hence, at points ( 1,2 ) and ( 1,−2 ), tangents are parallel to the x-axis.