Tangent of a Curve y =f(x)

Q.

The equation of tangent to the circle $x^2+y^2+2gx+2fy+c=0$ at its point $(x_1,y_1)$ is given by $x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c=0$

1. True

2. False

Q.

The slope of the tangent of the curve $y^2{e}^{xy}=9e^{-3}{x}^2$ at $\left(-1,3\right)$ is

1. $\frac{-15}{2}$

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

3. $15$

4. $\frac{15}{2}$

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

Q. If the line $3x-4y=0$ is tangent in the first quadrant to the curve $y=x^3+k,$ then value of $k$ is

1. $\frac{1}{2}$
2. $\frac{1}{4}$
3. $\frac{-1}{8}$
4. $\frac{-1}{2}$

Q. The number of points in the rectangle $\left\{\right(x,y)|-12\le x\le 12\text{and}-3\le y\le 3\}$ which lie on the curve $y=x+\sin x$ and at which the tangent to the curve is parallel to the $x$ - axis is

1. $0$
2. $2$
3. $4$
4. $8$

Q.

Number of possible tangents to the curve $y=cos(x+y),-3\pi \le x\le 3\pi$ that are parallel to the line x+2y = 0, is

1. 1

2. 2

3. 3

4. 4

Q.

The curve $y-e^{xy}+x=0$ has a vertical tangent at the point

1. (1, 1)

2. No point

3. (0, 1)

4. (1, 0)

Q.

What is the condition for a line $y=mx+c$ to be a tangent to the parabola$(y-k{)}^2=4a(x-h)$.

1. 

2. 

3. 

4. 

Q. The tangent at a point $P$ on the curve $x^m{y}^n=a^{m+n}$ meets the $x$-axis at $A$ and $y$-axis at $B$. The triangle formed by $AOB$ has centroid at $Q$, If $O$ be the origin, then

1. $AP:PB=n:m$
2. $AP:PB=m:n$
3. locus of $Q$ is $\left(3xm{)}^m\right(3yn{)}^n=\left[a\right(m+n){]}^{m+n}$
4. locus of $Q$ is $\left(3xm{)}^n\right(3yn{)}^m=\left[a\right(m+n){]}^{m+n}$

Q.

The angle between the tangents to the curve $y^2=2ax$ at the points where $x=\frac{a}{2}$, is

1. $\frac{\pi }{6}$

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

3. $\frac{\pi }{3}$

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

Q.

Find the equation of the tangent to the circle $x^2+y^2=a^{2}at(acos\alpha ,asin\alpha )$.

1. 

2. 

3. 

4. 

Q. The equations of the tangents to the ellipse 9x² + 16y² = 144 from the point (2, 3) are
(a) y = 3, x = 5
(b) x = 2, y = 3
(c) x = 3, y = 2
(d) x + y = 5, y = 3

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. -1, 2

3. 1, 2

4. 2, 7

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. 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. \N
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q. Find the slope of the tangent to the following curve y²=8(x-6) at the point (8, -4)?

Q.

Find the equation of tangent at $x=0$ on the curve $y=x{e}^x$

1. $y+x=0$

2. $y-x=0$

3. $y+\ln 2x=0$

4. $y-\ln 2x=0$

Q. find equation of tangent to the urve y(x3-1)(x-2) at the points where the curve curve cuts at x axis

Q.

The slope of the tangent to the curves $x=t^2+3t-8$, $y=2t^2-2t-5$ at the point $\left(2,-1\right)$ is

1. $\frac{22}{7}$

2. $\frac{6}{7}$

3. $-6$

4. $-7$

Q. If m is slope of common tangents y = $x^2$ - x + 1, y = $x^2$ - 3x + 1 then m is

1. 16
2. 7
3. 9
4. -2

Q. If the tangent to the curve $x={\mathrm{at}}^2,y=2\mathrm{at}$ is perpendicular to x-axis, then its point of contact is .

1. (a, a)
2. (0, a)
3. (0, 0)
4. (a, 0)

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

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

Q. Which of the following is (are) the coordinate(s) of the points on the curve $y=x^2+3x+4$, the tangents at which pass through the origin?

1. (2, 14)
2. (2, -14)
3. (2, 2)
4. (-2, 2)