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. $\frac{1}{4}$
• C. 2
• D. $\frac{1}{2}$

Answer 1

The correct option is A 4

$y=x^3\Rightarrow \frac{dy}{dx}=3x^2$ let $A(x_1,x_1^3),B(x_2,x_2^3)$
$m_A=3x_1^2=\frac{x_2^3-x_1^3}{x_2-x_1}=x_2^2+x_1{x}_2+x_1^2$
$\Rightarrow x_2^2+x_1{x}_2-2x_1^2=0\Rightarrow (x_2-x_1)(x_2+2x_1)=0$
$\therefore x_2=-2x_1(\because x_1\ne x_2)$
$K=\frac{m_B}{m_A}=\frac{3x_2^2}{3x_1^2}=4$