Question

The number of tangents to the curve $x^{3/2}+y^{3/2}=2a^{3/2}$, $a>0$, which are equally inclined to the axes, is

• A. $2$
• B. $1$
• C. $0$
• D. $4$

Answer 1

Answer: (B)

$1$

For a tangent to be equally inclined to the axes, the slope of the tangent should be 1.

Thus

$y^{\prime }=1$.

Now

$x^{\frac{3}{2}}+y^{\frac{3}{2}}=2a^{\frac{3}{2}}$.

$\frac{3}{2}\left(\sqrt{x}\right)+\frac{3}{2}\left(\sqrt{y}\right)=0$
$\sqrt{x}=-\sqrt{y}$

$x=y$

$x_1=y_1$

Substituting int he above equation, we get

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

$x=y=a$.

Hence the tangent touches the curve at $(a,a)$.

Thus we get only one tangent, satisfying that criteria.