Question

The number of points on the curve $x^{3/2}+y^{3/2}=a^{3/2}$, where the tangents are equally inclined to the axes, is

• A. $2$
• B. $1$
• C. $0$
• D. $4$

Answer 1

The correct option is B $1$

If the tangents are equally inclined to the axes we get $y_{x,y}^{\prime }=1$.

Therefore

$\frac{3}{2}\sqrt{x}+\frac{3}{2}\sqrt{y}.y^{\prime }=0$

$y^{\prime }=1$ implies

$\sqrt{x}+\sqrt{y}=0$

$\sqrt{x}=-\sqrt{y}$

$x=y$.

$x_1=y_1$.

Substituting in the equation we get

$2x^{\frac{3}{2}}=a^{\frac{3}{2}}$

$x=\frac{a}{2^{\frac{2}{3}}}$

Therefore

$(x_1,y_1)=(\frac{a}{2^{\frac{2}{3}}},\frac{a}{2^{\frac{2}{3}}})$

Therefore there exists only one point.