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

Q. The equation of tangent to curve $y=e^{-x}$ at the points where the curve cuts the line x = 1 is

1. e (x + y) = 1
2. x+ y = e
3. y + ex = 1
4. ey+x=2

Q. The angle at which curve $y=k{e}^{kx}$ intersects the y axis is

1. $ta{n}^{-1}\left(k^2\right)$
2. $co{t}^{-1}\left(k^2\right)$
3. $si{n}^{-1}\frac{1}{k^2}$
4. None of these

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

Q. If y = 4x - 5 is a tangent to the curve $y^2$ = $p{x}^3$ + q at (2, 3) then :

1. p = -2, q = 7
2. p =2, q = 7
3. p = 2, q = -7
4. p = -2, q = -7

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

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

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. The tangent to the curve x = a$\sqrt{cos2\theta }cos\theta ,y=a\sqrt{cos2\theta }sin\theta$ at the point corresponding to $\theta =\frac{\pi }{6}$ is

1. parallel to the x-axis
2. None of these
3. parallel to the y-axis
4. parallel to line y = x

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. $\frac{3}{2}$

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

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

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