Tangent and Normal

Show that the curves $x=y^2$ and $xy=k$ cut at right angles, if $8k^2=1$.

Show that the condition that the curves $a{x}^2+b{y}^2=1$and $a\text{'}{x}^2+b\text{'}{y}^2=1$ should intersect orthogonally is that $\frac{1}{a}-\frac{1}{b}=\frac{1}{a\text{'}}-\frac{1}{b\text{'}}$.

The equation of the tangent to the curve $f \left(x\right)=1-e^{\frac{x}{2}}$ at the point of intersection with the $y$-axis, is

Find the equation of the tangent to the curve

$x=a\left(\theta +\sin \theta \right), y=a\left(1+\cos \theta \right) at \theta =\frac{\mathrm{\pi }}{2}$

Find the equation of tangent and normal to the curve at the indicated point on it.

$2x^2+3y^2-5=0 \text{at} \left(1,1\right)$

Find the equations of tangent and normal to the curves at the indicated point on it.

$y=x^2+4x+1 \text{at} \left(-1,-2\right)$

The equation of the tangent to the curve $y=x^2-2x+7$ which is perpendicular to the line $5y-15x=13$ is

Find the equation of the normal to the curve $y=\sqrt{x-3}$ which is parallel to the line $6x+3y-4=0$

Find the equations of the tangent and the  normal to the following curves.

$y=x^3-x^2-1$ at the point whose abscissa is $-2$

Find the equations of the tangent and the normal to the following curves.

$x=a \sec \theta , y=a \tan \theta$ at $\theta =\frac{\mathrm{\pi }}{6}$

Find the equations of the tangent and the normal to the following curve.

$x=\frac{1}{t}, y=t-\frac{1}{t}$ at $t=2$

Find the equations of the tangent and the normal to the following curve.

$y=\sqrt{2}\sin \left(2x+\frac{\mathrm{\pi }}{4}\right)$ at $x=\frac{\mathrm{\pi }}{4}$

Find the equations of the tangent and the normal to the following curve.

$\sqrt{\mathrm{x}}-\sqrt{\mathrm{y}}=1 \mathrm{at} \mathrm{P}\left(9,4\right)$

Find the equations of the tangent and the normal to the following curve.

$y=x^2+4x$ at the point whose ordinate is $-3$

Find the equations of the tangent and the normal to the following curve.

$xy=c^2$ at $\left(ct,\frac{c}{t}\right)$ where $t$ is a parameter

Find the equations of the tangent and the normal to the curve $x^2+y^2+xy=3$ at $\mathrm{P}\left(1,1\right)$

The points on the curve $9y^2=x^3$, where the normal to the curve makes equal intercepts with the axes are

Coordinates of a point on the curve $y=x\log x$ at which the normal is parallel to the line $2x-2y=3$, are

The normal at the point $\left(1,1\right)$ on the curve $2y+x^2=3$ is

The normal to the curve $x^2=4y$ passing through $\left(1,2\right)$ is