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Important Questions on Application of Derivatives

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The line $y=x+1$ is a tangent to the curve $y^2=4x$ at the point

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The tangent to the curve, $y=x{e}^{x^2}$ passing through the point $\left(1,e\right)$ also passes through the point:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The length of the perpendicular from the origin, on normal to the curve, $\mathrm{}{x}^2+2xy-3y^2=0$, at the point $\left(2,2\right)$, is.

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Consider the curve $y=b{e}^{-x/a}$, where $a$ and $b$ are non-zero real numbers. Then

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The tangent at the point $(2,-2)$ to the curve, $x^2{y}^2-2x=4\left(1-y\right)$ does not pass through the point:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

A tangent to the curve $x=a{t}^2, y=2at$ is perpendicular to $x$-axis, then the point of contact is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent to the curve $y=\frac{x}{x^2-3}, x\in R, \left(x\ne \pm \sqrt{3}\right),$ at a point $\left(\alpha , \beta \right)\ne \left(0, 0\right)$ on it is parallel to the line $2x+6y-11=0,$ then:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent to the curve $xy+ax+by=0$ at $\left(1, 1\right)$ makes an angle $\tan {}^{-1}2$ with $x$-axis, then $\frac{a+b}{ab}=$

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

Let C be a curve given by $y\left(x\right)=1+ \sqrt{4x-3}$,$x>\frac{3}{4}$. If $P$ is a point on C, such that the tangent at $P$ has slope $\frac{2}{3}$, then a point through which the normal at $P$ passes, is :

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangent at a point $\mathrm{P}$, with parameter $t$, on the curve $x=4t^2+3, y=8t^3-1, t \in R,$ meets the curve again at a point $\mathrm{Q}$, then the coordinates of $\mathrm{Q}$ are :

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the normal to the curve $y=e^x$  at $\left(0, 1\right)$ is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The normal to the curve  $y\left(x-2\right) \left(x-3\right)=x+6$ at the point where the curve intersects the $y$-axis passes through the point:

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The point at which the tangent line to the curve of $y=\frac{16}{x}-x^2$ is horizontal, is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the curves $2x=y^2$ and $2xy=K$ intersect perpendicularly, then the value of $K^2$ is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the tangent to the curve $y=\pi e^{\frac{-x}{\pi }}$ at the point where it crosses $y$-axis is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the two curves $x=y^2$ and $xy=k$ cut each other at right angles, then a possible value of $k$ is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the tangent to the curve $x{y}^5+2x^{2}y-x^3+y+1=0$ at $x=0$ is

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

If the tangents on the ellipse $4x^2+y^2=8$ at the points $\left(\mathrm{1,} 2\right)$ and $(a, b)$ are perpendicular to each other, then $a^2$ is equal to

MEDIUM

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The point on the curve $y= \sqrt{x-1}$ where the tangent is perpendicular to the line $2x+y-5=0$ is

EASY

Mathematics>Differential Calculus>Application of Derivatives>Tangents and Normals

The equation of the tangent and normal to the ellipse $x^2+2y^2+2x-4y-14=0$ at $\left(2,-1\right)$ is