Normal of a Curve y = f(x)

Q. The equation of the normal to the curve $y=\sin \frac{\pi x}{2}$ at $(1,1)$ is

1. y = 1
2. x = 1
3. $y-1=\frac{-1}{\pi }(x-1)$
4. y = x

Q. If $M(x_0,y_0)$ is the point on the curve $3x^2-4y^2=72,$ which is nearest to the line $3x+2y+1=0,$ then the value of $(x_0+y_0)$ is equal to

1. $3$
2. $-3$
3. $9$
4. $-9$

Q.

Find the point on the curve $y=x^3-11x+5$ at which the tangent is y=x-11

Q. The slope of the tangent to a curve $y=f\left(x\right)$ at $(x,f(x\left)\right)$ is $2x+1.$ If the curve passes through the point $(1,2)$ then the area of the region by the curve, the $x$ -axis and the line $x=1$ is

1. $\frac{5}{6}$
2. $\frac{6}{5}$
3. $\frac{1}{6}$
4. $1$

Q.

Find the area of the smaller region bounded by the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and the line $\frac{x}{a}+\frac{y}{b}=1.$

Q. Let a curve $y=y\left(x\right)$ be given by the solution of the differential equation
$\cos \left(\frac{1}{2}{\cos }^{-1}\left(e^{-x}\right)\right)dx=\sqrt{e^{2x}-1}dy$
If it intersects $y-$ axis at $y=–1,$ and the intersection point of the curve with $x-$ axis is $(\alpha ,0),$ then $e^{\alpha }$ is equal to:_____.

Q.

The equation of the tangent of the curve ${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=2$ at $(a,b)$ is

1. $\left[\frac{x}{a}\right]+\left[\frac{y}{b}\right]=2$

2. $\left[\frac{x}{a}\right]+\left[\frac{y}{b}\right]=\frac{1}{2}$

3. $\left[\frac{x}{a}\right]-\left[\frac{y}{b}\right]=2$

4. $ax+by=2$

Q. The $x$-intercept of the tangent to a curve is twice the ordinate of the point of contact. The equation of the curve through the point $(1,1)$ is

1. $x=e^{\frac{2x-y}{2x}}$
2. $y=e^{\frac{y-x}{2y}}$
3. $y=e^{\frac{2y-x}{4x}}$
4. $x=e^{\frac{x-y}{2x}}$

Q.

Find the equations of tangents and normal at the extremities of latus rectum of the parabola $y^2=4ax.$

1. Equation of tangent is y = x + a

2. Equation of tangent is y = -x - a

3. Equation of normal is y = -x + 3a

4. Equation of normal is y = x - 3a

Q.

Find the equation of the tangent and normals to the curve $y=x^3+2x+6$ which are parallel to the line x+14y+4=0.

Q. If the line $ax+by+c=0$ is a normal to curve $xy+5=0,$ then $a$ and $b$ have

1. Same sign
2. Opposite sign
3. Can't be discussed
4. None of these

Q.

Find the area enclosed between the parabola $y^2=4ax$ and the line y = mx.

Q. The angle of intersection of the normals at the point $(-\frac{5}{\sqrt{2}},\frac{3}{\sqrt{2}})$ of the curves $x^2-y^2=8$ and $9x^2+25y^2=225$ is

1. $\frac{\pi }{2}$
2. $\frac{\pi }{3}$
3. $\frac{\pi }{4}$
4. $0$

Q.

Find points at which the tangent to the curve $y=x^3-3x^2-9x+7$ is parallel to the x-axis.

Q. Given the curve ${\left(\frac{x^2-y^2}{xy}\right)}^3=\frac{512}{27},$ then

1. the slope of normal drawn to the curve at $(3,1)$ is $\frac{1}{3}$
2. the value of
   ${(2\frac{dy}{dx}-5\frac{d^{2}y}{d{x}^2})}_{(3,1)}=\frac{2}{3}$
3. the slope of normal drawn to the curve at $(3,1)$ is $-3$
4. the equation of tangent at $(3,1)$ to given curve $3y+x=0$.

Q. Show that the function $f\left(x\right)=\{\begin{array}{cc}|2x-3|\left[x\right],& x\ge 1\\ \sin \left(\frac{\pi x}{2}\right),& x<1\end{array}$ is continuous but not differentiable at $x=1$

Q.

Find the equation of the lines having slope -1 that are tangent to the curve $y=\frac{1}{x-1},x\ne 1$

Q. The normal to the curve x 2 = 4 y passing (1, 2) is (A) x + y = 3 (B) x − y = 3 (C) x + y = 1 (D) x − y = 1

Q. The equation of the normal at the point (a, b) to the curve $y=4x^2$ is x + 24y - 864 = 0. Then, a = and b =

1. 3
2. 6
3. 36
4. 144

Q. Find the length of sub -tangent of parabola $y^2=8x$ at $(2,4)$.

1. None of these
2. $9$
3. $4$
4. $8$

Q. The differential equation satisfied by the curves $e^{2y}+2bx{e}^y+b^2=0,$ where $b$ is a parameter, is

1. $(1+x^2)\frac{d^{2}y}{d{x}^2}=0$
2. $(x^2-1){\left(\frac{dy}{dx}\right)}^2-1=0$
3. $(x^2-1)\frac{d^{2}y}{d{x}^2}+2=0$
4. $(x^2-x)\frac{dy}{dx}-y^2=0$

Q. There are two perpendicular straight lines touching the parabola $y^2=4a(x+a)$ and $y^2=4b(x+b)$, then the point of intersection of these two lines lie on the line given by

1. $x+a+b=0$
2. $x-a-b=0$
3. $x-a+b=0$
4. $x+a-b=0$

Q. A tangent is drawn to the curve $y=x^3+3x^2+1$ at $(h,k);h>0$. If the equation of tangent is $y=9x-4$ then the value of $h+k$ is

1. $5$
2. $6$
3. $8$
4. $4$

Q. Let the tangent at point $P(8,4)$ to the curve $y^3=x^2$ meets the curve again at $Q$ and $PQ=a\sqrt{10}.$ Then the value of $a$ is

Q.

Find the equation of the tangent and normal to the given curve at the given points

$y=x^{3}at(1,1)$

Q. Find the equations of the tangent and the normal, to the curve $16x^2+9y^2=145$ at the point $(x_1,y_1)$, where $x_1=2$ and $y_1>0$.

Q. Find the equation of the normal to the curve $y=x^3+2x+6$ which are parallel to the line $x+14y+4=0$.

Q. The point of the curve $y^2=2(x-3)$ at which the normal is parallel to the line y - 2x + 1 = 0 is
[MP PET 1998]

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

Q.

Equation of normal to the curve at y = $4x^2$ at (a, b) is x+16y-258 = 0 . Find the value of (a+b)

___

Q. If $x=t^2$and y = 2t, then equation of the normal at t = 1 is
[RPET 1996]

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