Tangents & normals practice · Mixed difficulty

Equation of normal to $x{\text{ = }}2{e^t},y = {e^{ - t}}{\text{ }}at{\text{ }}t = 0$ is

If the function $f(x)=x^2+\alpha / x$ has a local minimum at $x=2$, then the value of $\alpha$ is

The gradient of the curve passing through $(4,0)$ is given by $\frac{dy}{dx}-\frac{y}{x}+\frac{5x}{\left( x+2 \right)\left( x-3 \right)}=0$, if the point $(5,a)$ lies on the curve, then the value of $a$ is

If the sum of the squares of the intercepts on the axes cut off by the tangent to the curve ${x^{1/3}} + {y^{1/3}} = {a^{1/3}}$ (with $a > 0$) at $P\left( {a/8,\;a/8} \right)$ is 2, then $a =$

If the line $y=\sqrt{2} x-4 k \sqrt{2}$ is normal chord of parabola $y^2=2 x$ whose length is $a \sqrt{a}(a \in N)$, then the value of $\left(a+\frac{1}{k}\right)$ is

The equation of the tangent to the curve $y=(2 x-1) e^{2(1-x)}$ at the point of its maximum, is

The angle between the curves ${y^2} = 4x + 4$ and ${y^2} = 36\left( {9 - x} \right)$ is

If the curve $y=a x^2+b x+c$ passes through the point $(1,2)$ such that the slope of the tangent at the origin is equal to 1, then $(a, b, c)=$

If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3 y^2=9$ at the points $(3 \cos \theta, \sqrt{3} \sin \theta)$ and $(-3 \sin \theta, \sqrt{3} \cos \theta) ; \in\left(0, \frac{\pi}{2}\right) ;$ then $\frac{2 \cot \beta}{\sin 2 \theta}$ is equal to

The angle between the curves $y = {a^x}$ and $y = {b^x}$ is equal to

Let y be the number of people in a village at time $t$. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increase in time. Then, the population of the village at any fixed $t$ is given by

The curves $x=y^2$ and $x y=a^3$ cut orthogonally at a point, then $a=$

If the tangent at the point $(1,1)$ of the curve $y^2-x(2-x)^2$ meets the curve again at the point $\mathrm{P}$, then $\mathrm{P}$ is ...... .

The area of a triangle formed by a tangent to the curve $2 x y=a^2$ and the coordinate axes, is

The curve $y=a x^3+b x^2+c x+5$ touches the $\mathrm{x}$-axis at $\mathrm{P}(-2,0)$ and cut the $\mathrm{y}$-axis at a point $\mathrm{Q}$ where its gradient is 3 . Then, $a+b+c=$

If the curve $y = 2{x^3} + a{x^2} + bx + c$ passes through the origin and the tangents drawn to it at $x = - 1$ and $x = 2$ are parallel to the $x$-axis, then the values of $a,b$ and care respectively

The point on the curve $y = 2{x^2} - 6x - 4$ at which the tangent is parallel to the $x$-axis, is

The equation of the tangent to the curve $x = t\cos t,\;y = t\sin t$ at the origin is

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

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

Point of contact of tangents to the curve $y^{2}-2 x^{3}-4 y+8=0$ from the point $(1,2)$ is/are :

If the curves $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{12}} = 1\;$ and ${y^3} = 8x\;$ intersect at right angle then the value of ${a^2}$ is equal to

Which one of the following curves is the orthogonal trajectory of straight lines passing through a fixed point $(a, b)$ ?

The point of inflexion for the curve $y = {x^{5/2}}$ is

The number of points on the curve $y = {x^3} - 2{x^2} + x - 2$ where tangents are parallel to $x$-axis, is

The curve $y=x^{\frac{1}{5}}$ at $(0,0)$ has

The point in the interval $\;\left[ {0\;,2\pi } \right]$, where $f\left( x \right) = {e^x}\;\sin x\;$ has maximum slope, is

The equation of the one of the tangents to the curve $y = \cos \left( {x + y} \right),\; - 2\pi \le x \le 2\pi$ that is parallel to the line $x + 2y = 0$, is

The intercepts made by the tangent to the curve $y = \int \limits_0^x \left| t \right|dt$, which is parallel to the line $y = 2x$, on $y$-axis are equal to

The subtangent at any point on the curve $x^m y^n=a^{m+n}$ varies as