Equation of Tangent at a Point (x,y) in Terms of f'(x)

Q. the point of intersection of the straight line 3x+5y=1 and (2+c)x+5c^2y=1 where c not equal to 1 and -3/2, is

Q.

If the minor axis of an ellipse forms an equilateral triangle with one vertex of the ellipse then e =

1. 

2. 

3. 

4. 

Q. 19. the angle x whose cosine is equal to its tangent given by sinx = nsin18. then n is equal to =

Q.

If the circles $x^2+y^2+2x+2ky+6=0,x^2+y^2+2ky+k=0$ intersect orthogonally, then k is

1. $2or-\frac{3}{2}$

2. $-2or\frac{3}{2}$

3. $2or-\frac{3}{2}$

4. $-2or\frac{3}{2}$

Q. The slope of the tangent to the curve$(y-x^5{)}^2=x(1+x^2{)}^2$ at the point $(1,3)$ is ___

Q.

The equation of the tangent to the curve $y=1-e^{x/2}$ at the point of intersection with the y- axis is

1. x +2y =0

2. 2x +y =0

3. x - y =2

4. 2x+y=0

Q. If the tangent at (1, 1) on $y^2=x(2-x{)}^2$ meets the curve again at P, then P is

1. (4, 4)
2. (-1, 2)
3. $(\frac{9}{4},\frac{3}{8})$
4. (0, 0)

Q. The straight lines x+y = 0, 3x+y-4 = 0, and x+3y-4 = 0 form a triangle which is
a. isosceles
b. equilateral
c. right angled
d. none of these.

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

1. $(\frac{-7}{17},\frac{6}{17})$
2. $(\frac{-6}{17},\frac{10}{17})$
3. $(\frac{-4}{17},\frac{1}{17})$
4. $(\frac{-8}{17},\frac{2}{17})$

Q.

Find the equation of a line passing through (1, 2) and making angle of 30⁰ with y-axis.

Q.

The tangent to the curve $x^2+y^2=25$ parallel to the line 3x-4y=7 exist at the point

1. (-3, -4)

2. (3, -4)

3. (3, 4)

4. (1, 1)

Q. find the equation of common tangent y2-6y-4x+9=0 and x2+y2-6x-6y+9=0

Q.

The triangle formed by the tangent to the curve $f\left(x\right)=x^2+bx-b$at the point $(1,1)$ and the coordinate axes lie in the first quadrant. If its area is $2$, then the value of $b$ is

1. $-1$

2. $3$

3. $-3$

4. $1$

Q. The equation of a line passing through $(-2,3)$ and parallel to the tangent at origin for circle $x^2+y^2+x-y=0$ is:

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

Q. The y - intercept of tangent drawn to the curve $x=t^2+3t-8$ and $y=2t^2-2t-5$ at the point $(2,-1)$ is

1. $\frac{12}{7}$
2. $\frac{15}{7}$
3. $\frac{-11}{7}$
4. $\frac{-19}{7}$

Q. If ‘P’ be a point on the graph of $y=\frac{x}{1+x^2}$ then co-ordinates of ‘P’ such that tangent drawn to the curve at ‘P’ has greatest slope in magnitude is

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

Q. The coordinates of the point P on the curve $y^2=2x^3$, at which the tangent is perpendicular to the line 4x - 3 y + 2 = 0, are given by

1. (2, 4)
2. $(1,\sqrt{2})$
3. $(\frac{1}{2},-\frac{1}{2})$
4. $(\frac{1}{8},-\frac{1}{16})$

Q.

The portion of the tangent at any point on the curve $x=a{t}^3$, $y=a{t}^4$ between the axes is divided by the abscissa of the point of contact externally in ratio

1. $\frac{1}{4}$

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

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

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

Q. Let $x^{2/3}+y^{2/3}=a^{2/3},(a>0)$ be the equation of curve then which of the following is/are correct?

1. Equation of tangent at $(x_1,y_1)$ on the curve is $\frac{x}{x_1^{1/3}}+\frac{y}{y_1^{1/3}}=a^{2/3}$
2. Length of portion of tangent intercepted between the coordinate axes is constant.
3. If the curve $x^{2/3}+y^{2/3}=a^{2/3}$ touches the curve $\frac{x^2}{c^2}+\frac{y^2}{d^2}=1$ then $c+d=a$
4. Equation of normal at $(x_1,y_1)$ on the curve is $x.x_1^{1/3}-y.y_1^{1/3}=x_1^{4/3}+y_1^{4/3}$
   

Q. The tangent to the curve given by, $x=e^t.\cos t,y=e^t.\sin t\mathrm{at}t=\frac{\pi }{4},$ makes an angle of with x-axis.

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