Interaction between Two Curves

Find the equation of tangent of the curve $x=\theta +\sin \theta , y=1+\cos \theta$ at $\theta =\frac{\mathrm{\pi }}{4}$.

Find the equation of tangent of the curve $x=\frac{2a{t}^2}{1+t^2}, y=\frac{2a{t}^3}{1+t^2}$ at $t=\frac{1}{2}$.

Find the equation of tangent of the curve $x=\theta +\sin \theta , y=1+\cos \theta$ at $\theta =\frac{\mathrm{\pi }}{2}$.

Find the equation of the tangent at the point $\text{'}t\text{'}$ on the curve $x=a{\sin }^{3}t, y=b{\cos }^{3}t$.

Find the equation of the tangent line to the curve $x=1-\cos \theta , y=\theta -\sin \theta$ at $\theta =\frac{\mathrm{\pi }}{4}$.

Consider the two curves $C_1: y^2=4x$ and $C_2: x^2+y^2-6x+1=0$, then

The angle of intersection between the curves $x^2=4(y+1)$ and $x^2=-4(y+1)$ is

The acute angle between the curves $y=x^2$ and $x=y^2$, at there common point other than origin, is

Consider the two curves $C_1: y^2=4x$ and $C_2: x^2+y^2-6x+1=0$, then

The angle of intersection of the curves $x^2=4y$ and $y^2=4x$ at a point other than origin is,

Note: You may use $\cos 53°=\frac{3}{5}$

Find the condition that the curves $2x=y^2$ and $2xy=k$ intersect orthogonally.

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

Find the angle of intersection between the two curves $C_1:$$y=3^x$ and  $C_2:$ $y=4^x$, is equal to:

The angle $\theta , (\theta <\frac{\mathrm{\pi }}{2})$  between the curves $\mathrm{y}=\left|x^2-1\right|$ and $\mathrm{y}=\left|x^2-3\right|$ at their points of intersection is-

Possible value(s) of $a$ for which curves ${\mathrm{C}}_1;\mathrm{y}=\sqrt{3}+\mathrm{sinx},\mathrm{x}\in (0,2\pi )$ and ${\mathrm{C}}_2;\mathrm{y}=\frac{\left|\mathrm{x}\right|}{2}+\mathrm{a},$ have a common tangent, is / are :

The number of values of $a$ for which the curves $4x^2+a^2{y}^2=4a^2$ and $y^2=16x$ are orthogonal is

The cosine of the acute angle between the curves $y=\left|x^2-1\right|$ and $y=\left|x^2-3\right|$ at their points of intersection is

The acute angle of intersection of the curves $x^{2}y=1$ and $\mathrm{}y=x^2$ in the first quadrant is $\mathrm{}\theta ,$ then $\tan \theta$ is equal to

If the curves C₁ : 3x² + 4y² = 12 and C₂ : 12x² – cy² – 3 = 0 are orthogonal to each other then c =

If $\theta$ denotes the acute angle between the curves, $y=10-x^2$ and $y=2+x^2$ at a point of their intersection, then $\left|\tan \theta \right|$ is equal to: