The area of the triangle formed by the tangent to the curve $xy = a^2$ at $(x_1, y_1)$ on it and the coordinate axes is:

Find the point on the curve $x = a(\theta + \sin \theta)$,$y = a(1 - \cos \theta)$ where the tangent makes an angle of $\pi / 4$ with the $x$-axis.

The equation of the tangent to the curve $y = \cos(x + y)$ for $-2\pi \leq x \leq 2\pi$,which is parallel to the line $x + 2y = 0$,is:

For the curve $4x^{5} = 5y^{4}$,the ratio of the cube of the subtangent at a point on the curve to the square of the subnormal at the same point is

Find the equations of the tangent and normal to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=2$ at $(1,1).$

Let $f: R \rightarrow R$ be a bijection. $A$ curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0$ for all $x \in R$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cut the $X$-axis at $A$ and $B$ respectively,and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $AC+CB$ is minimum,then the tangent at $P$ is parallel to the line