Question

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

Answer 1

Tangents are parallel to the x - axis.
so, slope of tangent = 0 -----(1)

given, x² + y² - 2x- 3 = 0
differentiate with respect to x,
2x + 2y.dy/dx - 2 = 0
(x - 1) + y.dy/dx = 0
dy/dx = (1 - x)/y
Let (a,b) is the point on the curve at which the tangents are parallel to the x - axis.
at (a, b) , dy/dx = (1 - a)/b -----(2)

from equations (1) and (2),
0 = (1 - a)/b
=> a = 1

put (a, b) in the curve a² + b² - 2a - 3 = 0 ---(3)

put a = 1 in equation (3),
1 + b² - 2 - 3 = 0
=> b² - 4 = 0
=> (b - 2)(b + 2) = 0
=> b = ± 2

hence , required points (1,2) and (1,-2)