Let $m$ be the slope of the normal $L$ drawn at $(1, 2)$ to the curve $x = t^2 - 7t + 7, y = t^2 - 4t - 10$ and $ax + by + c = 0$ be the equation of the normal $L$. If the $G$.$C$.$D$. of $(a, b, c)$ is $1$, then $m(a + b + c) =$

What is the length of the normal at any point on the curve $y = \frac{1}{2}a(e^{x/a} + e^{-x/a})$?

Let $f(x)$ be a differentiable function,$A(0, \alpha)$ and $B(8, \beta)$ be two points on the curve $y=f(x)$. Given $f(0)=2$ and $f^{\prime}(4)=\frac{-3}{4}$. If the chord $AB$ of the curve is parallel to the tangent drawn at the point $(4, f(4))$,then $\beta=$

If the tangent to the curve $y = f(x)$ at a point is perpendicular to the line $2x - 3y = 5$,then the value of $\frac{dy}{dx}$ at that point is:

If the line $x + By + C = 0$ is the normal to the curve given by $x = a \sin^3 t$ and $y = b \cos^3 t$ (where $a, b \neq 0$) at the point $t = \frac{\pi}{2}$, then $B - C =$

The area of the triangle formed by the tangent and the normal drawn to the curve $y^2=4x$ at $(1,2)$ with the $Y$-axis is (in square units):