Question

The number of tangents to 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 $1$

$\frac{dy}{dx}=-1$ (where x is equally inclined)

$\frac{3}{2}{x}^{{}^1{/}_2}{+}^3{/}_2{y}^{{}^1{/}_2}\frac{dy}{dx}=0$

$\frac{dy}{dx}=-\sqrt{\frac{x}{y}}=1$

$-\sqrt{\frac{x}{y}}=-1$

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

$x=y$

$x^{{}^3{/}_2}+x^{{}^3{/}_2}=a^{{}^3{/}_2}$

$2x^{{}^3{/}_2}=a^{{}^3{/}_2}$

$x=\frac{a}{2^{{}^2{/}_3}}$

$y=\frac{a}{2^{{}^2{/}_3}}$

$\therefore$ only one tangent.