Question
The number of tangents to the curve $${x^{\frac{3}{2}}} + {y^{\frac{3}{2}}} = 2{a^{\frac{3}{2}}},\,a > 0,$$ which are equally inclined to the axes, is :
A.
2
B.
1
C.
0
D.
4
Answer :
1
Solution :
Given the curve is $${x^{\frac{3}{2}}} + {y^{\frac{3}{2}}} = 2{a^{\frac{3}{2}}}.....(1)$$
$$\therefore \,\frac{3}{2}\sqrt x + \frac{3}{2}\sqrt y \frac{{dy}}{{dx}} = 0\,\,{\text{or, }}\frac{{dy}}{{dx}} = - \frac{{\sqrt x }}{{\sqrt y }}$$
Since the tangent is equally inclined to the axes,
$$\eqalign{
& \frac{{dy}}{{dx}} = \pm 1 \cr
& \therefore \, - \frac{{\sqrt x }}{{\sqrt y }} = \pm 1{\text{ or }} - \frac{{\sqrt x }}{{\sqrt y }} = - 1 \cr
& \therefore \,\sqrt x = \sqrt y \,\,\,\,\,\left[ {\because \,\sqrt x > 0,\,\sqrt y > 0} \right] \cr
& {\text{Putting }}\sqrt y = \sqrt x {\text{ in equation}}\left( 1 \right){\text{,we get}} \cr
& 2{x^{\frac{3}{2}}} = 2{a^{\frac{3}{2}}}{\text{ or }}{x^3} = {a^3} \cr
& {\text{Therefore, }}x = a{\text{ and, so }}y = a \cr} $$