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Tangent of a Curve y =f(x)

If the tangen...

Question

If the tangent to the curve $y = x^{3}$ at the point $P (t, t^{3})$ meets the curve again at $Q$ , then the ordinate of the point which divides $P Q$ internally in the ratio $1 : 2$ is :

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Solution

Equation of tangent at $P (t, t^{3})$
$(y - t^{3}) = 3 t^{2} (x - t)$
Now solve the above equation with
$y = x^{3}$
We have
$x^{3} - t^{3} = 3 t^{2} (x - t)$
$\Rightarrow (x - t) (x^{2} + x t + t^{2}) = 3 t^{2} (x - t)$
$\Rightarrow x^{2} + x t - 2 t^{2} = 0$
$\Rightarrow (x - t) (x + 2 t) = 0$
$\Rightarrow x = - 2 t \Rightarrow Q (- 2 t, - 8 t^{3})$
Ordinate of required point
$= \frac{2 t^{3} + (- 8 t^{3})}{3} = - 2 t^{3}$

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Tangent of a Curve y =f(x)

Standard XII Mathematics

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