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:

Let the quadratic curve passing through the point $\left(-1,0\right)$ and touching the line $y=x$ at $\left(1,1\right)$ be $y=f\left(x\right)$. Then the $x$-intercept of the normal to the curve at the point $\left(\alpha ,\alpha +1\right)$ in the first quadrant is

Find the slope of the normal to the curve $2x^2-xy+3y^2=18$ at $\left(3, 1\right)$.

A curve has equation $y=-3x^3+x-4$. The tangent to this curve at the point where $x=1$ meets the $x$-axis at the point $A$ and $y$-axis at the point $B$. Find the area of the triangle $OAB$.

Show that the equation of the normal to the hyperbola $x=asec\theta , y=b\tan \theta$ at the point $\left(asec\theta , b\tan \theta \right)$ is $ax\cos \theta +bycot\theta =a^2+b^2$.

Find the value of $k$ if $y=2x+k$ is normal to $x^2+1=y$.

If the normal at $\left(1,2\right)$ on the parabola $y^2=4x$ meets the parabola again at $\left(t^2,2t\right)$, then the value of $t$ is

Find the point where the normal at $\left(4,4\right)$ meets the parabola $y^2=4x$.

Find the equation of tangent and the normal to the parabola $y^2=4x$ at point $\left(4,-4\right).$

The value(s) of $a$ for which the area of triangle included between the axes and any tangent to the curve $x^{a}y={\lambda }^a$ is constant, is/are

Find common tangent between the curves $y=x^3$ and $112x^2+y^2=112.$

FInd the equation of normal passing through $\left(1,2\right)$ to the curve $x^2=4y.$

Consider the curve $f\left(x\right)=x^{\frac{1}{3}}$, then

Find the interval for $\alpha$ so that $\left(3-\alpha \right)x+\alpha y+\left({\alpha }^2-1\right)=0$ is normal to the curve $xy=4$.

The slope of the normal to the curve $y^3-xy-8=0$ at the point $\left(0,2\right)$ is equal to 

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 $k=$

If $x+13y=40$ is normal to the curve $y=5x^2+\alpha x+\beta$ at the point $\left(1,3\right)$, then the value of $\alpha \beta$ is equal to

The normal to the curve $y=\sqrt{x}$ at the point $\left(25,5\right)$ intersects the $y-$axis at

If tangent to the curve $xy=x^2+1$ at $\left(\alpha ,\beta \right)$ is normal to the curve $x^2+y^2+2fy+c=0$, then $\frac{1}{\left|f.\alpha \right|}$ is

The equation of tangent to the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at the point $\left(a,b\right)$ is $?$