A curve is such that the x -intercept of the tangent drawn to it at the point P x , y is reciprocal of the abscissa of P . Then, the equation of the curve is (where, c is the constant of integration and x 1 )

Question

A curve is such that the x -intercept of the tangent drawn to it at the point P x , y is reciprocal of the abscissa of P . Then, the equation of the curve is (where, c is the constant of integration and x > 1 ), y = c x 2 - 1, y = c x 2 + 1, y = c x 2 - 1, y = c x - 1

MARKS App · Accepted Answer

The general equation of the tangent is Y - y = d y d x X - x ∴ x -intercept = x - y m ⇒ x - y m = 1 x ⇒ x - 1 x = y m Or m = y x - 1 x ⇒ d y d x = y x - 1 x ⇒ d y y = x d x x 2 - 1 On integrating, we get ∫ d y y = ∫ x x 2 - 1 d x ⇒ ln ⁡ y = 1 2 ∫ 2 x d x x 2 - 1 = 1 2 ln ⁡ x 2 - 1 + ln ⁡ c ⇒ ln ⁡ y = ln ⁡ x 2 - 1 c ⇒ y = c ⋅ x 2 − 1

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