Question

If the tangent to the curve \(y^{2}=x^{3}\) at \(\left(m^{2}, m^{3}\right)\) is also a normal to the curve at \(\left(M^{2}, M^{3}\right)\), then the value of \(m M\) is
(A) \(-\frac{1}{9}\)
(B) \(-\frac{2}{9}\)
(C) \(-\frac{1}{3}\)
(D) \(-\frac{4}{9}\)

Answer 1

Ans : (D)
Hint : \(2 y y_{1}=3 x^{2}\)
$$
y_{1}=\frac{3 x^{2}}{2 y} \Rightarrow\left(y_{1}\right)_{m^{2}, m^{3}}=\frac{3 \times m^{4}}{2 \times m^{3}}=\frac{3 m}{2}
$$
Again ; slope of normal \(=-\frac{2}{3 M}, \mathrm{mM}=-\frac{4}{9}\)