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

The equation of the given curve is x2 + y2 − 2x − 3 = 0.

On differentiating with respect to x, we have:

2x + 2y $\frac {dy}{dx}$ - 2 = 0

y$\frac {dy}{dx}$ = 1 - x

$\frac {dy}{dx}$ = $\frac {1-x}{y}$

Now, the tangents are parallel to the x-axis if the slope of the tangent is 0.

$\frac {1-x}{y}$= 0 $\implies$1-x = 0 $\implies$x = 1

But, x2 + y2 − 2x − 3 = 0 for x = 1.

y2 = 4

$\implies$y2=4$\implies$y=±2

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