Tangent and Normal

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 a line $px+qy=12$ is a tangent to the ellipse $4x^2+9y^2=16$, then $pq$ can not be equal to :

The number of parallel tangents of $f\left(x\right)=x^2-x+1$ and $g\left(x\right)=x^3-x^2-2x+1$ is

The tangent to a given curve is perpendicular to $x$-axis if

The tangent to the curve $3x{y}^2-2x^{2}y=1$ at $(1,1)$ meets the curve again at the point

Find the slope of tangent to the curve $y=2x^2+3\sin x$ at $x=0$.

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

The points on the curve $\frac{x^2}{4}+\frac{y^2}{25}=1$ at which the tangent is parallel to $x$-axis are 

Find the slope of the tangent to the curve $y=x^3-x$ at $x=2$.

Equations of tangents to the ellipse $2x^2+3y^2=11$ at the points whose ordinates is $1$

The equation of the normal to the curve $2x^2-y^2=14$ which are parallel to the line $x+3y=8$ 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 tangent of a curve $y=x^2-2x+3$ which is parallel to the line $2x-y+9=0$ is 

If the line $6x-y-4=0$ touches the curve $y^2=a{x}^3+b$ at the point $\left(1,2\right)$, then $a+b=$

Let $f:\mathrm{ℝ}\to \mathrm{ℝ}$ be a bijection. A curve represented by $y=f\left(x\right)$ is such that $f^{\text{'}}\left(x\right)>0\forall x\in \mathrm{ℝ}.$ The tangent and normal drawn at $P(\alpha ,1)$ on the curve cuts the $X$ -axis at $A,B$ respectively and $C$ is the foot of the perpendicular from $P$ onto the $X$ -axis. If $P(\alpha ,1)$ is such a point that $AC+CB$ is minimum, then the tangent at $P$ is parallel to the line

The equation of the normal to the curve $x=a\cosh \left(t\right), y=b\sinh \left(t\right)$ at any point $t$ is

The curve $y=a{x}^3+b{x}^2+cx+5$ touches the $x$-axis at $P\left(-2,0\right)$ and cuts $y$-axis at a point $Q$, where gradient is $3.$ Then the values of $a, b, c$ are

If the tangent to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ meets the $x$-axis at $A$ and $y$-axis at $B,$ then $AB=$