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

Q. Let $E$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus at $(4,-4)$ and one vertex at $(5,-4)$. If $mx-y=4,m>0$ is a tangent to the ellipse $E$, then the value of $5m^2$ is equal to

Q.

Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect a point P in the first quadrant. If the normal to this ellipse at P meets the co-ordinate axes at $\left(-\frac{1}{3\sqrt{2}},0\right)$ and $\left(0,\beta \right)$, then $\beta$ is equal to

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

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

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

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

Q. Let $P(x_0,y_0)$ be a point on the curve $C:(x^2-11)(y+1)+4=0,$ where $x_0,y_0\in N.$ If area of the triangle formed by the normal drawn to the curve $C$ at $P$ and the co-ordinate axes is $\frac{a}{b},$ where $a,b\in N,$ then the least value of $a-6b$ is

Q. Let the normal at a point $P$ on the curve $y^2-3x^2+y+10=0$ intersects the $y$-axis at $(0,\frac{3}{2})$. If $m$ is the slope of the tangent at $P$ to the curve, them $|m|$ is equal to

Q. The equation of the normal to the curve $y=2x^2+3\sin x$ at x = 0 is .

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

Q. If the normal of $y=f\left(x\right)$ at $(0,0)$ is given by $y-x=0,$ then
$\underset{x\to 0}{lim}\frac{x^2}{f\left(x^2\right)-20f\left(9x^2\right)+2f\left(99x^2\right)}$

1. equals $\frac{1}{2}$
2. equals $\frac{1}{19}$
3. equals $\frac{-1}{19}$
4. does not exist

Q.

What is the parametric form of the equation of the ellipse given by,

x²/a² + y²/b² = 1

Q.

For the straight lines 4x + 3y – 6 = 0 and 5x + 12y + 9 = 0, the equation of the bisector of the angle which contains the origin is

1. 7x + 9y + 3 = 0

2. 7x – 9y + 3 = 0

3. 7x + 9y – 3 = 0

4. None of these

Q.

e^(sin x) - e^(-sin x) = 4 then find the number of real solutions

Q. The distance, from the origin, of the normal to the curve, $x=2\cos t+2t\sin t,y=2\sin t–2t\cos t,\text{at}t=\frac{\pi }{4}$, is:

1. $4$
2. $\sqrt{2}$
3. $2$
4. $2\sqrt{2}$

Q. Let $S$ be the area bounded by the curve $y=\sin x,0\le x\le \pi$ and the $x-$axis and $T$ be the area bounded by the curves $y=\sin x,0\le x\le \frac{\pi }{2},$ $y=a\cos x,0\le x\le \frac{\pi }{2}$ and the $x-$axis, where $a\in R^{+}$. If $S:T=1:\frac{1}{3},$ then which of the following is(are) CORRECT ?

1. The value of $a$ is $\frac{4}{3}$
2. The value of $S$ is equal to $1$.
3. The value of $T$ is equal to $\frac{2}{3}$
4. The value of $a$ is $\frac{2}{3}$

Q. Let $C$ be a curve given by $y\left(x\right)=1+\sqrt{4x–3},x>\frac{3}{4}$. If $P$ is a point on $C$, such that the tangent at $P$ has slope $\frac{2}{3}$, then a point through which the normal at $P$ passes, is:

1. $(3,-4)$
2. $(1,7)$
3. $(2,3)$
4. $(4,-3)$

Q. The abscissa of a point on the curve $xy=(a+x{)}^2,$ at which the normal cuts off numerically equal intercepts from the coordinate axes is

1. $-\frac{a}{\sqrt{2}}$
   
2. $\sqrt{2}a$
   
3. $\frac{a}{\sqrt{2}}$
   
4. $-\sqrt{2}a$

Q. The area of the region bounded by the curves $|x+y|\le 2,|x-y|\le 2$ and $2x^2+6y^2\ge 3$ is

1. $(8+\frac{\sqrt{3}}{2}\pi )sq.units$
   
2. $(8-\frac{\sqrt{3}}{2}\pi )sq.units$
   
3. $(4-\frac{3\sqrt{3}}{2}\pi )sq.units$
   
4. $(8-\frac{3\sqrt{3}}{2}\pi )sq.units$

Q. The abscissa of the point on the curve $\sqrt{xy}=a+x,$ the tangent at which cuts off equal intercepts from the coordinate axes is

1. $\frac{a}{\sqrt{2}}$
2. $\frac{-a}{\sqrt{2}}$
3. $-a\sqrt{2}$
4. $a\sqrt{2}$

Q. Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $(-\frac{1}{3\sqrt{2}},0)$ and $(0,\beta )$, then $\beta$ is equal to:

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

Q. Let $C(m,n)$ be a point on the curve $y=x^3,$ where the tangent is parallel to the chord connecting the points $O(0,0)$ and $A(2,8)$. Then the value of $m\cdot n$ is:

1. $\frac{16}{9}$
2. $\frac{12}{5}$
3. $\frac{16}{27}$
4. $\frac{32}{9}$

Q. Which of the following statements is (are) INCORRECT about $f\left(x\right)=\frac{e^x}{x}$?

1. $f\left(x\right)$ is increasing in $(0,\infty )$.
2. Tangent on the curve $y=f\left(x\right)$ at $x=1$ is perpendicular to $x-$axis.
3. Tangent on the curve $y=f\left(x\right)$ at $x=1$ is parallel to $x-$axis.
4. Range of $f\left(x\right)$ is $[e,\infty )$.

Q. 36. dy/dx+2xtan(x-y)=1 then sin(x-y) (1) origin (2) the line x=1/2 (3)The point (1, 0) (4) the point (1/2, 0)

Q. The triangle formed by the normal to the curve $f\left(x\right)=x^2-ax+2a$ at the point $(2,4)$ and the coordinate axes lie in second quadrant. If its area is $2$ sq.units, then the sum of possible values of $a$ is

Q. The equation of the normal to the curve $y=2x^2+3\sin x$ at x = 0 is .

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

Q. The area enclosed by the curve $y=\sin x+\cos x$ and $y=|\cos x-\sin x|$ over the interval $[0,\frac{\pi }{2}]$ is

1. $4(\sqrt{2}-1)$
2. $2\sqrt{2}(\sqrt{2}-1)$
3. $2(\sqrt{2}+1)$
4. $2\sqrt{2}(\sqrt{2}+1)$

Q. For the curve $x=t^2-1,y=t^2-t$, tangent is parallel to $x$ - axis where,

1. $t=0$
2. $t=\frac{1}{2}$
3. $t=-\frac{1}{\sqrt{3}}$
4. $t=\frac{1}{\sqrt{3}}$

Q. The equation of the plane through the intersection of the planes x + 2y + 3z = 4 and 2x + y − z = −5 and perpendicular to the plane 5x + 3y + 6z + 8 = 0 is

(a) 7x − 2y + 3z + 81 = 0
(b) 23x + 14y − 9z + 48 = 0
(c) 51x − 15y − 50z + 173 = 0
(d) None of these

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

Q. The centre of a circle of radius $4\sqrt{5}$ lies on the line y=x and satisfies the inequality 3x+6y>10. If the line x+2y=3 is a tangent to the circle, then the equation of the circle is

1. 
2. 
3. 
4. 

Q. Let $\left[k\right]$ denote the greatest integer less than or equal to $k.$ If $S$ is the area enclosed by the curves $f\left(x\right)=4|x|-|x{|}^3$ and $g\left(x\right)+\sqrt{4-x^2}=0,$ then the value of $\left[S\right]$ is

Q. If the line aX + bY + c =0 is a normal to the curve xy =1. Then

1. $a>0,b>0$
2. $a>0,b=0$
3. $a<0,b>0$
4. $a<0,b<0$

Q. If normal to the curve y = f(x) is parallel to x-axis, then correct statement is [RPET 2000]

1. $\frac{dy}{dx}=0$
2. $\frac{dy}{dx}=1$
3. $\frac{dx}{dy}=0$
4. None of these

Q. If $A$ is $(x_1,y_1)$ where $x_1=1$ on the curve $y=x^2+x+10$, the tangent at $A$ cuts the x-axis at $B$. The value of $\overline{OA}.\overline{AB}$ is

1. $148$
2. $-352$
3. $-148$
4. $140$