Tangent and Normal

The equations of the normal to the curve  $x=1-\cos \theta :y=\theta -\sin \theta$ at $\theta =\frac{\pi }{4}$  would be:

The equation of a tangent to the curve $y=\sqrt{3x-2}$ which is parallel to the line $4x-2y+5=0$ would be

If the straight line $x\cos \alpha +y\sin \alpha =p$ touches the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at the point $\left(a,b\right)$ on it and $\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$ then $k=$

The number of those tangents to the curve $y^2-2x^3-4y+8=0$ which pass through the point $\left(1,2\right)$ is

$f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e, a\ne 0$, is a function such that $x=1, x=2$ and $x=3$ are normals to the curve $y=f\left(x\right)$ such that $f\left(2\right)$ is always greater than $f\left(0\right)$. If $f\left(x\right)=e-a^2-10a+12$, has no real solutions, then the number of all possible integral values of $a$ is

The points on the curve $y=\frac{2}{3}{x}^3+\frac{1}{2}{x}^2$, where the tangents make equal angles with coordinate axes are

The point on the curve $x^2+y^2=a^2, y\ge 0, a>0$ at which the tangent is parallel to the $x$-axis, is

If the relation $p$(subnormal length)$=q(\mathrm{subtangent} \mathrm{length}{)}^2$ holds true for the curve $b{y}^2=$ $(x+a{)}^3,$ then the value of $\frac{p}{q}=$_________

Find the coordinates of a point on the curve $y=x^2-3\mathrm{x}+2,$ at which the tangent drawn to this curve is perpendicular to the line $y=x.$

Find the area of the triangle formed by the $x$-axis and the tangent and the normal to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $\left(\frac{a}{\sqrt{2}}, \frac{b}{\sqrt{2}}\right)$

The sum of the lengths of the subtangent and the subnormal drawn at $\theta =\frac{\pi }{3}$ on the cycloid $x=a(\theta -\sin \theta ); y=a(1-\cos \theta )$ is $\_\_\_\_\_\_\_\_\_$

The point on the curve $y=x^3,$ at which the tangent to the curve is parallel to the $x$-axis, is _____

If the tangent to the curve $2{\mathrm{y}}^3=a{x}^2+x^3$ at the point $(a,a)$ cuts off intercepts $\alpha$ and $\beta$ on the co-ordinate axes, where ${\alpha }^2+{\beta }^2=61,$ then the value of $\left|a\right|$ is $\_\_\_\_\_\_\_\_$

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

If the parabola $y=\alpha x^2-6x+\beta$ passes through the point $\left(0, 2\right)$ and has its tangent at $x=\frac{3}{2}$ parallel to $x$ axis, then

Let $P\left(a\sec \theta ,b\tan \theta \right)$ and $Q\left(a\sec \varphi ,b\tan \varphi \right)$ where $\theta +\varphi =\frac{\pi }{2},$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If $\left(h,k\right)$ is the point of intersection of normals at $P$ and $Q,$ then $k$ is equal to

If the normal to the curve $y=f\left(x\right)$ at the point $\left(3,4\right)$ makes an angle $\frac{3\pi }{4}$ with the positive $x$-axis, then $f^{\prime }(3)$ is equal to

The tangent at the point$(2,-2)$ to the curve, $x^2{y}^2-2x=4\left(1-y\right)$ does not pass through the point:

Find the gradient of the following curve at the point where $x=3$,

$g\left(x\right)=\left(x^2-3\right)(x-1{)}^5$.

Find the gradient of the following curve at the point where $x=3$,

$y=x{e}^{2x}$[ Write approximate value correcting to only numerical value, use $e^6=403.428793$]