If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{2/3} + y^{2/3} = 4$ is parallel to the line $\sqrt{3}x + y = 1$,then $\alpha^2 + \beta^2 =$

If the equation of the normal to the curve $y = \frac{x-a}{(x+b)(x-2)}$ at the point $(1, -3)$ is $x - 4y = 13$,then the value of $a+b$ is equal to $.......$.

The sum of the intercepts on the axes of the tangent to the curve $\sqrt{x} + \sqrt{y} = \sqrt{a}$ is equal to:

Show that the equation of the normal at any point on the curve $x=3 \cos \theta-\cos ^{3} \theta, y=3 \sin \theta-\sin ^{3} \theta$ is $4(y \cos ^{3} \theta-x \sin ^{3} \theta)=3 \sin 4 \theta$.

The abscissa of the point on the curve $y=a\left(e^{\frac{x}{a}}+e^{-\frac{x}{a}}\right)$ where the tangent is parallel to the $X$-axis is

If $(a^2-1) x+a y+(3-a)=0$ is a normal to the curve $x y=1$,then the interval in which '$a$' lies is