Question

If the tangent at A on the curve y=$x^3$ meets the curve again at B and the gradient at B is K times the gradient at A, then the value of K is

• A. 4
• B.
• C. 2
• D.

Answer 1

The correct option is A

4

y = $x^3$ $\Rightarrow$ $\frac{dy}{dx}$ = 3$x^2$ let A $(x_1,x_1^3),B(x_2,x_2^3)$

$m_A=3x_1^2=\frac{x_2^3-x_1^3}{x_2-x_1}=x_2^2+x_1{x}_2+x_1^2$

$\Rightarrow x_2^2+x_1{x}_2-2x_1^2=0$

$\Rightarrow (x_2-x_1)(x_2+2x_1)=0$

$\therefore$ $x_2=-2x_1(\because x_1\ne x_2)$

K= $\frac{m_B}{m_A}$ = $\frac{3x_2^2}{3x_1^2}$ = 4