Question

lf the tangent at any point on the curve $x^4+y^4=c^4$ cuts off intercepts $a$ and $b$ on the coordinate axes, the value of $a^{-4/3}+b^{-4/3}$ is

• A. $c^{-4/3}$
• B. $c^{-1/2}$
• C. $c^{1/2}$
• D. $c^{4/3}$

Answer 1

The correct option is A $c^{-4/3}$

Given equation of curve is
$x^4+y^4=c^4$
$\frac{dy}{dx}=-\frac{x^3}{y^3}$
Slope of tangent at point $(x_1,y_1)$ on the curve is $-\frac{x_1^3}{y_1^3}$
Let the equation of tangent to curve be
$\frac{x}{a}+\frac{y}{b}=1$
Since, this line intersects X-axis at $(a,0)$ and $(0,b)$
$\frac{x}{c^4/x_1^3}+\frac{y}{c^4/y_1^3}=1$
$\Rightarrow a=\frac{c^4}{x_1^3},b=\frac{c^4}{y_1^3}$

$\Rightarrow a^{-4/3}=\frac{c^{-16/3}}{x_1^{-4}},b^{-4/3}=\frac{c^{-16/3}}{y_1^{-4}}$

$\Rightarrow a^{-4/3}+b^{-4/3}=c^{-16/3}(x_1^4+y_1^4)$
$\Rightarrow a^{-4/3}+b^{-4/3}=c^{-4/3}$