Tangent of a Curve y =f(x)

Q.

At what point on the curve $x^3-8a^{2}y=0,$ the slope of the normal is $-\frac{2}{3}$.

1. $\left(a,a\right)$

2. $\left(2a,-a\right)$

3. $\left(2a,a\right)$

4. none of these

Q. If $ta{n}^{-1}\left(\frac{x-2}{x-4}\right)+ta{n}^{-1}\left(\frac{x+2}{x+4}\right)=\frac{\pi }{4}$, find the value of x.

Q.

The equation of the tangent to the curve $x=2{\cos }^3\left(\theta \right)$ and $y=3{\sin }^3\left(\theta \right)$ at the point, $\theta =\frac{\mathrm{\pi }}{4}$ is

1. $2x+3y=3\sqrt{2}$

2. $2x-3y=3\sqrt{2}$

3. $3x+2y=3\sqrt{2}$

4. $3x-2y=3\sqrt{2}$

Q. The equation of the circle passing through the point (–1, –3) and touching the line 4x + 3y – 12 = 0 at the point (3, 0), is

1. 
2. 
3. 2
4. None of these

Q.

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

1. $(0,2),(2,16)$
2. $(2,16),(–2,–16)$
3. $(2,–16),(–2,16)$
4. $(0,–2),(2,–16)$

Q. If the curves $y=\frac{\ln x}{x}$ and $y=\lambda x^2$ (where $\lambda$ is constant) touch each other, then $\lambda$ is -

1. $\frac{e}{3}$
2. $\frac{1}{3e}$
3. $\frac{3}{e}$
4. $3e$

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

1. $\frac{x}{a}+\frac{y}{b}=1$ is tangent to the curve at $(0,0)$.
2. $\frac{x}{a}+\frac{y}{b}=1$ is tangent to the curve where the curve crosses the axis of $y$.
3. $\frac{x}{a}+\frac{y}{b}=1$ is tangent to the curve at $(a,0)$
4. $\frac{x}{a}+\frac{y}{b}=1$ is tangent to the curve at $(2a,0).$

Q.

Chose the correct answer.

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

(a) (1, 2) (b) (2, 1) (c) (1, -2) (d) (-1, 2).

Q.

Tangents are drawn from the origin to the curve y = sin x. Their points of contact lie on the curve

1. x²y²=y+x

2. x²y²=x²-y²

3. None of these

4. x²y²=x²+y²

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

1. $(2,3e)$
2. $(3,6e)$
3. $(\frac{4}{3},2e)$
4. $(\frac{5}{3},2e)$

Q. Find points on the curve at which the tangents are (i) parallel to x -axis (ii) parallel to y -axis

Q. A curve is represented parametrically by $x=t+e^{at}$ and $y=-t+e^{at},t\in R$ and $a>0$. Then, the curve touches the $x-$ axis at

1. $(1,0)$
2. $(\frac{1}{e},0)$
3. $(e,0)$
4. $(2e,0)$

Q. If tangents are drawn to the curve $y=\frac{x}{1+x^2}$ such that the tangents are concurrent at $(1,\frac{1}{2}),$ then

1. Number of such tangents is $3$.
2. Each of the tangents has exactly two points in common with the curve.
3. Abscissa of point of contact for one of the tangents is $1+\sqrt{2}$.
4. Abscissa of point of contact for one of the tangents is $1-\sqrt{2}$.

Q.

If the line y = 2x touches the curve y = ax^2 + bx + c at the point where x = 1 and the curve passes through the point (-1, 0), then the values of a, b, c are

1. 
   

2. 
   

3. 
   

4. None of these

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

1. $|6\alpha +2\beta |=19$
2. $|2\alpha +6\beta |=11$
3. $|6\alpha +2\beta |=9$
4. $|2\alpha +6\beta |=19$

Q. The tangent to the curve $y=e^{kx}$ at the point $(0,1)$ meets the $x-$axis at $(a,0)$ where $a\in [-2,-1]$, then $k\in$

1. $[-\frac{1}{2},0]$
2. $[-1,-\frac{1}{2}]$
3. $[0,1]$
4. $[\frac{1}{2},1]$

Q. If the area of the triangle included between the axes and any tangent to the curve $x^{n}y=a^n$ is constant, then the value of $n$ is

Q. The distance between the origin and the tangent to the curve $y=e^{2x}+x^2$ drawn at the point $x=0$ is

1. $\frac{1}{\sqrt{5}}$
2. $\frac{2}{\sqrt{5}}$
3. $\frac{-1}{\sqrt{5}}$
4. $\frac{2}{\sqrt{3}}$

Q.

The point on the curve $y=x^3$ at which the tangent to the curve is parallel to the $x$ axis is

1. $(2,2)$

2. $(3,3)$

3. $(4,4)$

4. $\left(0,0\right)$

Q.

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

1. 1, 2

2. 2, 7

3. 1, -2

4. -1, 2

Q. $Lety=4x^2$ & $\frac{x^2}{a^2}-\frac{y^2}{16}=1$ intersect

1. a>\sqrt{2}
2. \left | a \right |> \frac{1}{\sqrt{2}}
3. 
4. \left | a \right |>-\frac{1}{\sqrt{2}}

Q.

The chord joining the points where x =p and x=q on the curve $y=a{x}^2+bx+c$ is parallel to the tangent at the point on the curve whose abscissa is

1. $\frac{p+q}{2}$

2. $\frac{p-q}{2}$

3. $\frac{pq}{2}$

4. $p+q$

Q.

If a curve passes through the origin and the slope of the tangent to it at any point $(x,y)$ is $\frac{x^2-4x+y+8}{x-2}$, then this curve also passes through the point :

1. $(4,5)$

2. $(5,4)$

3. $(4,4)$

4. $(5,5)$

Q.

Find the slope of the normal to the curve $4x^3-3x{y}^2+6x^2-5xy-8y^2+9x+14=0$ at the point $(-2,3)$.

1. $\infty$

2. $1$

3. $\frac{9}{2}$

4. $\frac{2}{9}$

Q. At what point, the slope of the tangent to the curve $x^2+y^2-2x-3=0$ is zero :

1. (1, 2) (1, -2)
2. (3, 0) ( 1, 2)
3. (3, 0); (-1, 0)
4. (-1, 0) (1, 2)

Q. For the curve y = 3sin $\theta cos\theta ,x=e^{\theta }sin\theta ,0\le \theta \le \pi$; tangent is parallel to x-axis, then $\theta$ =

1. 0
2. 
3. 
4. 

Q. If $\theta$ is angle between curves $y=[|\sin x|+|\cos x|]$, $\left(\right[\cdot \left]\text{denotes greatest integer function}\right)$ and $x^2+y^2=5$ then $4{\text{cosec}}^2\theta$ is​​​​​​​

Q.

A horse runs along a circle with a speed of 20 km/hr. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers $\frac{1}{8}$ of the circle in km/hr is

1. 20

2. 40

3. 30

4. 60

Q. The equation of the normal to the curve $y(1+x^2)=2-x$ where the tangent crosses $x$-axis is

1. $5x-y-10=0$
2. $x-5y-10=0$
3. $5x+y+10=0$
4. $x+5y+10=0$

Q.

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x⁴ − 6x³ + 13x² − 10x + 5 at (0, 5)

(ii) y = x⁴ − 6x³ + 13x² − 10x + 5 at (1, 3)

(iii) y = x³ at (1, 1)

(iv) y = x² at (0, 0)

(v) x = cos t, y = sin t at