Application of Derivatives

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

Question

Find the points on the curve x² + y² -2x – 3 = 0 at which the tangents are parallel to the x-axis.

Solution

The equation of curve is x² + y² -2x – 3 = 0 ...(1)
Differentiating both sides, w.r.t.x, we get,
$2 straight x plus 2 straight y dy over dx minus 2 minus 0 space equals space 0 comma space space space space or space space space 2 straight y dy over dx space equals space minus 2 straight x plus 2 therefore space space space space space space dy over dx space equals space fraction numerator negative straight x plus 1 over denominator straight y end fraction because space space space space space dy over dx space equals space 0 space space space space rightwards double arrow space space space space space space fraction numerator negative straight x plus 1 over denominator straight y end fraction space equals space 0 space space space space rightwards double arrow space space space straight x space equals 1 comma space space space space straight y not equal to 0 Putting space straight x space equals space 1 space in space left parenthesis 1 right parenthesis comma space we space get comma space space space space space space space space space space space space space space space space space space space space space space 1 plus straight y squared minus 2 minus 3 space equals space 0 comma space space space space or space space space space straight y squared space equals space 4 space space space space rightwards double arrow space space space space straight y space equals space minus 2 comma space space space 2 therefore space space space space space space space space pionts space left parenthesis 1 comma space minus 2 right parenthesis comma space space left parenthesis 1 comma space 2 right parenthesis.$

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