Slope Form of Normal : Ellipse

Q. Number of distinct normals, that can be drawn to the ellipse $\frac{x^2}{169}+\frac{y^2}{25}=1$ from the point $P(0,6)$ is

1. one
2. two
3. three
4. four

Q. The tangent and normal to the ellipse $3x^2+5y^2=32$ at the point $P(2,2)$ meet the $x-$axis at $Q$ and $R$, respectively. Then the area (in sq. units) of the triangle $PQR$ is :

1. $\frac{34}{15}$
2. $\frac{16}{3}$
3. $\frac{14}{3}$
4. $\frac{68}{15}$

Q. What is the equation of the normal with slope m to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1?$

1. 

2. 

3. 

4. 

Q. A normal inclined at an angle of $\frac{\pi }{4}$ to the x-axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is drawn. It meets the major and minor axes in P and Q respectively. If C is the centre of the ellipse then the area of the triangle CPQ is

1. 
2. 
3. 
4. 

Q. The area bounded by $y^2=4x$ and the line y = 2x – 4 is

1. 9
2. 5
3. 4
4. 2

Q.

The area bounded by the curves $y=\sqrt{5-x^2},y=|x-1|$ is

1. $\left[\frac{5\mathrm{\pi }}{4}-2\right]$

2. $\left[\frac{5\mathrm{\pi }-2}{4}\right]$

3. $\left[\frac{5\mathrm{\pi }-2}{2}\right]$

4. $\left[\frac{\mathrm{\pi }}{2}-5\right]$

Q. The points of the ellipse $16x^2+9y^2=400$ at which the ordinate decreases at the same rate at which the abscissa increases is/are given by

1. $(3,\frac{16}{3})\&(-3,\frac{-16}{3})$
2. $(3,\frac{-16}{3})\&(-3,\frac{16}{3})$
3. $(\frac{1}{16},\frac{1}{9})\&(-\frac{1}{16},-\frac{1}{9})$
4. $(\frac{1}{16},-\frac{1}{9})\&(-\frac{1}{16},\frac{1}{9})$

Q. Coordinates of point(s) on the ellipse $x^2+3y^2=37,$ where the normal is parallel to the line $6x-5y=2,$ is/are

1. $(5,-2)$
2. $(5,2)$
3. $(-5,2)$
4. $(-5,-2)$

Q. Find the shortest distance between the lines whose vector equations are $\overrightarrow{r}=(\hat{i}+2\hat{j}+3\hat{k})+\lambda (\hat{i}-3\hat{j}+2\hat{k})$ and $\overrightarrow{r}=(4\hat{i}+5\hat{j}+6\hat{k})+\mu (2\hat{i}+3\hat{j}+\hat{k})$.

Q. Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,(a>b)$ be given ellipse, length of whose latus rectum is $10.$ If its eccentricity is the maximum value of the function, $\varphi \left(t\right)=\frac{5}{12}+t-t^2,\text{then}{a}^2+b^2$ is equal to

1. $135$
2. $116$
3. $126$
4. $145$

Q.

What is the equation of the normal which is perpendicular to 3x + 4y = 5 for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

1. 

2. 

3. 

4. 

Q. A point on the ellipse $x^2+3y^2=37$ where the normal is parallel to the line $6x-5y=2$ is

1. $(5,-2)$
2. $(5,2)$
3. $(-5,2)$
4. $(-5,-2)$

Q. The equation $\frac{7}{r}=5+3\cos \theta +4\sin \theta$ represents

1. a parabola
2. an ellipse
3. a rectangular hyperbola
4. a hyperbola

Q. Prove that
$f\left(x\right)=\{\begin{array}{c}\frac{x^2-25}{x-5},whenx\ne 5\\ 10,whenx=5\end{array}$ is continuous at $x=5$.

Q. Consider an infinite geometric series with first term, a and common ratio r. If its sum is 4 and the second terms , is $\frac{3}{4}$ , then a = $\_\_\_\_\_\_$ and r = $\_\_\_\_\_\_$.

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

Q. Express $Z=1+i\sqrt{3}$. in polar form.

Q. The tangent and normal to the ellipse $3x^2+5y^2=32$ at the point $P(2,2)$ meet the $x-$axis at $Q$ and $R$, respectively. Then the area (in sq. units) of the triangle $PQR$ is :

1. $\frac{34}{15}$
2. $\frac{16}{3}$
3. $\frac{14}{3}$
4. $\frac{68}{15}$

Q. If $\beta$ is one of the angles between the normals to the ellipse, $x^2+3y^2=9$ at the points $(3\cos \theta ,\sqrt{3}\sin \theta )$ and $(-3\sin \theta ,\sqrt{3}\cos \theta )$; $\theta \in (0,\frac{\pi }{2});$ then $\frac{2\cot \beta }{\sin 2\theta }$ is equal to :

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

Q. The area of the region bounded by the parabola $(y-2{)}^2=x-1$, the tangent to it at the point where the ordinate is $3$ and the $x-$axis is

Q. Let $a,b,c>R^{+}$ ( i.e. $a,b,c$ are positive real numbers) then the following system of equations in $x,y,z$
$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$, $\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ and $\frac{-x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ has

1. No solution
2. Unique solution
3. Infinitely many solution
4. Finitely many solution

Q. The focus of the parabola $y^2-x-2y+2=0$ is

1. $(1,1)$
2. $\left(\frac{5}{4},1\right)$
3. $\left(\frac{1}{4},0\right)$
4. none of these

Q. $li{m}_{x\to 0}\frac{(1+x{)}^4-1}{(1+x{)}^3-1}=..........$

1. $\frac{4x}{3}$
2. $4x^3$
3. $\frac{4}{3}$
4. $\frac{3x}{4}$

Q. If $f\left(x\right)=\{\begin{array}{c}\left[x\right]+[-x],x\ne 2\\ \lambda ,x=2\end{array}$, $f$ is continuous at $x=2$ then $\lambda$ is (where $[\cdot ]$ denotes greatest integer)-

1. $1$
2. $-1$
3. $2$
4. $0$

Q.

1. True
2. False

Q. Consider the equation $a{z}^2+z+1=0$ having purely imaginary root where $a$ = cos$\theta +i$ sin $\theta$, $i=\sqrt{-1}$ and function $f\left(x\right)=x^3-3x^2+3(1$ + cos $\theta )x+5$, then answer the following questions.
Number of roots of the equation cos $2\theta$ = cos $\theta$ in $[0,4\pi ]$ are

1. 2
2. 3
3. 4
4. 6

Q. The equation of latus rectum of a parabola is $x+y=8$ and the equation of the tangent at the vertex is $x+y=12$. Then length of the latus rectum is

1. $4\sqrt{2}$
2. $8\sqrt{2}$
3. $8$
4. $2\sqrt{2}$

Q. If $P$ and $P^{\prime }$ are the perpendiculars from the origin, upon the straight lines $x\sec \theta +y\csc \theta =a$ and $x\cos \theta -y\sin \theta =a\cos 2\theta ,$ then the value of $4P^2+{P^{\prime }}^2$ is

1. $-12a^2$
2. $5a^2$
3. $a^2$
4. $-5a^2$

Q. A normal inclined at an angle of $\frac{\pi }{4}$ to the x-axis of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is drawn. It meets the major and minor axes in P and Q respectively. If C is the centre of the ellipse then the area of the triangle CPQ is

1. $\frac{(a^2-b^2{)}^2}{4(a^2+b^2)}$
2. $\frac{(a^2-b^2{)}^2}{(a^2-b^2)}$
3. $\frac{(a^2-b^2{)}^2}{2(a^2+b^2)}$
4. $\frac{(a^2+b^2{)}^2}{2(a^2+b^2)}$

Q. If $a\in (-1,1)$, then roots of the quadratic equation $(a-1)x^2+ax+\sqrt{1-a^2}=0$ are

1. real
2. imaginary
3. both equal
4. none of these

Q. The straight lines given by the equations $x+y=2,x-2y=5and\frac{x}{3}+y=0$ are?

1. intersecting to make a right triangle.
2. parallel to each other.
3. intersecting to make an isosceles triangle.
4. concurrent