Question

The lines tangent to the curves $y^3-x^{2}y+5y-2x=0$ and $x^4-x^3{y}^2+5x+2y=0$ at the origin intersect at an angle $\theta$ equal to

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{2}$

Answer 1

The correct option is D $\frac{\pi }{2}$

Given curves are, $y^3-x^{2}y+5y-2x=0..\left(1\right)$ and $x^4-x^3{y}^2+5x+2y=0..\left(2\right)$

differentiating both w.r.t $x$
$3y^2\frac{dy}{dx}-x^2\frac{dy}{dx}-2xy+5\frac{dy}{dx}-2=0$

and

$4x^3-3x^2{y}^2-2x^3{y}^2\frac{dy}{dx}+5+2\frac{dy}{dx}=0$
Thus, by putting coordinates $(0,0)$ of origin for $(x,y)$ in above equation $\left(1\right)$, slope of tangent at origin to the first curve is $m_1=\frac{2}{5}$
and, by putting coordinates $(0,0)$ of origin for $(x,y)$ in equation $\left(2\right)$ above, slope of tangent at origin to the second curve is $m_2=-\frac{5}{2}$
Clearly $m_1.m_2=-1$
Hence both the lines are perpendicular.