Tangents & normals practice · Mixed difficulty

The line $\frac{x}{a} + \frac{y}{b} = 2$ touches the curve ${\left( {\frac{x}{a}} \right)^n} + {\left( {\frac{y}{b}} \right)^n} = 2$ at the point (a,b) for

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

Angle between ${y^2} = x$ and ${x^2} = y$ at the origin is

If the tangent to the conic, $y-6=x^2$ at $(2,10)$ touches the circle, $x^2+y^2+8 x-2 y=k$ (for some fixed $k$ ) at a point $(\alpha, \beta)$; then $(\alpha, \beta)$ 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

If the straight line joining the points $(0,3)$ and $(5,-2)$ is tangent to the curve $y=\frac{c}{x+1}$; then the value of $c$ is :

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

Angle between the tangents to the curve $y = {x^2} - 5x + 6$ at the point (2, 0) and (3, 0) 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

The equation of a normal to the curve, $\sin \mathrm{y}=\mathrm{x} \sin \left(\frac{\pi}{3}+y\right)$ at $\mathrm{x}=0$, is :

The points on the curve $y=12 x-x^3$ at which the gradient is zero are

Let slope of the tangent line to a curve at any point $P\,(x,\,y)$ be given by $\frac{{x{y^2} + y}}{x}$. If the curve intersects the line $x + 2y = 4$ at $x = - 2$, then the value of $y$, for which the point $(3,\,y)$ lies on the curve, is

The subtangent, ordinate and subnormal to the parabola ${y^2} = 4ax$ at a point (different from the origin) are in

If at each point on the curve $y=x^3-a x^2+x+1$ the tangent is inclined at an acute angle with the positive direction of $x$-axis, then

The normal to the curve represented parametrically by $x = a\left( {\cos \theta + \theta \sin \theta } \right)\,and\,y = a\left( {\sin \theta - \theta \cos \theta } \right)$ at any point $\theta$, is such that it

The curve $x+y-\ell \mathrm{n}(x+y)=2 x+5$ has a vertical tangent at the point $(\alpha, \beta)$. Then $\alpha+\beta$ is equal to

If the rate of change of sine of an angle $\theta$ is $k$, then the rate of change of its tangent is

If $y = 4x - 5$ is tangent to the curve ${y^2} = p{x^3} + q$ at (2, 3) then (p,q) is

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

The abscissa of the points, where the tangent to the curve $y = {x^3} - 3{x^2} - 9x + 5$ is parallel to $x$-axis, are

If $y = 4x - 5$ is a tangent to the curve ${y^2} = p{x^3} + q$ at (2, 3), then

The equation of the normal to the curve $y=\sin x$ at $(0,0)$, is

The equation of the tangent to the curve $y = 4x{e^x}\,\mathrm{at}\,\left( { - 1,\frac{{ - 4}}{e}} \right)$ is

A tangent is drawn at the point $M$ with abscissa unity on the curve $y=\sqrt{\left(5-x^{\frac{2}{3}}\right)^3}$. If length of segment of tangent intercepted between the co-ordinate axes is $\sqrt{{N}}$, then ${N}$ is equal to

Tangent is drawn to ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1\,at\,\left( {3\sqrt 3 \cos \theta ,\sin \theta } \right)\,(where\,\theta \in \left( {0,\;\pi /2} \right))$. Then the value of $\theta$ such that sum of intercepts on axes made by this tangent is minimum, is

A curve passes through the point $(2,0)$ and the slope of the tangent at any point $(x, y)$ is $x^2-2 x$ for all value of $x$. The point of maximum ordinate on the curve is