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

Q. If the area of the triangle formed by the positive $x-$axis, the normal and the tangent to the circle $(x-2{)}^2+(y-3{)}^2=25$ at the point $(5,7)$ is $A,$ then $24A$ is equal to

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

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. A function $y=f\left(x\right)$ has a second order derivative $f"\left(x\right)=6(x-1).$ If its graph passes through the point $(2,1)$ and at that point the tangent to the graph is $y=3x-5,$ then the function is

1. $(x+1{)}^2$
2. $(x-1{)}^3$
3. $(x+1{)}^3$
4. $(x-1{)}^2$

Q. If a variable tangent to the curve $x^{2}y=c^3$ makes intercepts $a,b$ on $x$ and $y-$axis respectively, then the value of $a^{2}b$ is

1. $27c^3$
2. $\frac{4}{27}{c}^3$
3. $\frac{27}{4}{c}^3$
4. $\frac{4}{9}{c}^3$

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 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 $x^2+y^2=25$ 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. parallel to the y-axis
3. parallel to line y = x
4. 3X-4Y+2=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)