Tangents & normals practice · Easy

The length of the normal to the curve $y=a\left(\frac{e^{-x / a}+e^{x / a}}{2}\right)$ at any point varies as the

The abscissae of the point on the curve $x y=(a+x)^2$, the normal at which cuts off numerically equal intercepts from the axes of coordinates, is

Let $g\left( x \right) = \;\left\{ {\begin{array}{cccccccccccccccccccc} {2e\;\;\;\;\;\;\;\;\;\;\;\;\;if\;x \le 1}\\ {\mathrm{\;log}\left( {x - 1} \right),\;\;\;if\;\;x > 1} \end{array}} \right.\;$ . The equation of the normal to $y = g\left( x \right)$ at the point (3,log 2), is

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

If the slope of the curve $y=\frac{a x}{b-x}$ at the point $(1,1)$ is 2, then

The equation of the tangent to the curve $y=1-e^{x / 2}$ at the point of intersection with the $y$-axis, is

The equation of the tangent to the curve $y = {e^{ - \left| x \right|}}$ at the point where the curve cuts the line $x = 1$ is

If the slope of the tangent to the curve $y = {e^x}\cos x$ is minimum at $x = a,\;0 \le a \le 2\;\pi$, then the value of $a$ is

The length of tangent, subtangent, normal and subnormal for the curve $y = {x^2} + x - 1$ at (1, 1) are $A,\;B,\;C\,\mathrm{and}\,D$ respectively, then their increasing order is

Equation of a common tangent (if it exists) to $x^2+y^2=2$ and $x^3 y^3=1$ is

If $ST$ and $SN$ are the lengths of the subtangent and the subnormal at the point $\theta = \frac{\pi }{2}$ on the curve $x = a\left( {\theta + \sin \theta } \right),\;y = a\left( {1 - \cos \theta } \right),\;a \ne 1$, then

The equation of tangent at $(-4,-4)$ on the curve $x^2=-4 y$ is

The slope of the tangent to the curve $\mathrm{y}=\int_0^{\mathrm{x}} \frac{\mathrm{dx}}{1+\mathrm{x}^3}$ at the point where $\mathrm{x}=1$ is -

The equation of normal to the curve ${x^2}y = {x^2} - 3x + 6$ at the point with abscissa $x = 3$ is

The intercept on $x$-axis made by tangents to the curve $y=\int\limits_{0}^x|t| d t, x \in R$, which are parallel to the line $y=2 x$, are equal to

If an equation of a tangent to the curve, $y-\cos (x+y),-1-1 \leq x \leq 1+\pi$, is $x+2 y=k$ then $k$ is equal to :

The angle between the curves $x y=2$ and $y^2=4 x$ at their point of intersection is

The points at which the tangents to the curve $a x^2+$ $2 h x y+b y^2=1$ is parallel to $y$-axis is

If the tangent at any point on the curve $x^4+y^4=c^4$ cuts off intercepts $a$ and $b$ on the coordinate axes, the value of $a^{-4 / 3}+b^{-4 / 3}$ is

The length of the normal to the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$, at $\theta=\pi / 2$, is

Any tangent to the curve $y = 2{x^5} + 4{x^3} + 7x + 9$

For the curve $x=t^2-1, y=t^2-t$, the tangent line is perpendicular to $x$-axis when

The tangent and the normal drawn to the curve $y = {x^2} - x + 4$ at $P\left( {1,\;4} \right)$ cut the $x$- axis at $A$ and $B$ respectively. If the length of the subtangent drawn to the curve at $P$ is equal to the length of the subnormal, then the area of the triangle $PAB$ in sq units is

The equation of the normal to the parabola, $x^2=8 y$ at $x=4$ is

The line $\frac{x}{a} + \frac{y}{b} = 1$ touches the curve $y = b{e^{ - x/a}}$ at the point

Let $y=f(x)$ be the equation of a parabola which is touched by the line $y=x$ at the point where $x=1$. Then,

The angle between the tangents to the curve $y=x^2-5 x+6$ at the point $(2,0)$ and $(3,0)$, is

If $y(\alpha ) = \sqrt {2\left( {\frac{{\tan \alpha + \cot \alpha }}{{1 + {{\tan }^2}\alpha }}} \right) + \frac{1}{{{{\sin }^2}\alpha }}} ,\alpha \in \left( {\frac{{3\mathrm{\pi }}}{4},\mathrm{\pi }} \right),$ then $\mathrm{ }\frac{{dy}}{{d\alpha }}$ at $\alpha = \frac{{5\mathrm{\pi }}}{6}$ is

The length of the normal at point '$t$' of the curve $x = a\left( {t + \sin t} \right),\;y = a\left( {1 - \cos t} \right)$ is

The equation of the normal to the curve ${y^4} = a{x^3}\;$ at (a, a) is