The tangent to the curve $y^2 - xy + 9 = 0$ is vertical when

The equation of the normal to the curve $\sin y = \sqrt{3} x \sin \left(\frac{\pi}{6} + y\right)$ at $x = 0$ is:

The slope of the normal to the curve $y=\frac{x}{x^2+1}$ at $x=-4$ is

If curves $\frac{x^2}{a^2} + \frac{y^2}{4} = 1$ and $y^3 = 16x$ intersect orthogonally,then $a^2$ equals:

Let $\Gamma$ be the curve $y=b e^{-x/a}$ and $L$ be the straight line $\frac{x}{a}+\frac{y}{b}=1$, where $a, b \in \mathbb{R}$. Which of the following statements is true?

The slope of a tangent drawn at the point $P(\alpha, \beta)$ lying on the curve $y=\frac{1}{2x-5}$ is $-2$. If $P$ lies in the fourth quadrant,then $\alpha-\beta=$