Directrix of Ellipse

Q. The equation of the ellipse with vertices $(\pm 13,0)$ and foci $(\pm 5,0),$ is

1. $\frac{x^2}{169}+\frac{y^2}{144}=1$
2. $\frac{x^2}{13}+\frac{y^2}{12}=1$
3. $\frac{x^2}{13}+\frac{y^2}{11}=1$
4. $\frac{x^2}{169}+\frac{y^2}{121}=1$

Q.

If the distance between the directrices of a rectangular hyperbola are $10$, then the distance between its foci will be

1. $10\sqrt{2}$

2. $5$

3. $5\sqrt{2}$

4. $20$

Q. Let $F_1(x_1,0)$ and $F_2(x_2,0),$ for $x_1<0$ and $x_2>0,$ be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1.$ Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.

The orthocentre of the triangle $F_{1}MN$ is

1. $(\frac{2}{3},0)$
2. $(\frac{9}{10},0)$
3. $(\frac{2}{3},\sqrt{6})$
4. $(-\frac{9}{10},0)$

Q. Let the normals at the four points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ and $(x_4,y_4)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be concurrent at some point (called as conormal point). Then $(x_1+x_2+x_3+x_4)(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4})$ is equal to

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

Q. If the distance between the directrices be thrice the distance between the foci, then eccentricity of ellipse is

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

Q. The equation of the ellipse with foci at $(\pm 5,0)$ and $x=\frac{36}{5}$ as one directrix, is

1. $\frac{x^2}{11}+\frac{y^2}{36}=1$
2. $\frac{x^2}{36}+\frac{y^2}{9}=1$
3. $\frac{x^2}{9}+\frac{y^2}{36}=1$
4. $\frac{x^2}{36}+\frac{y^2}{11}=1$

Q.

If the distance between a focus and corresponding directrix of an ellipse be 8 and the eccentricity be $\frac{1}{2}$ , then length of the minor axis is

1. 3

2. 

3. 6

4. None of these

Q. The foci of the ellipse, $25(x+1{)}^2+9(y+2{)}^2=225$ are

1. $(-1,2)$ and $(-1,-6)$
2. $(-1,-2)$ and $(-1,-6)$
3. $(-2,1)$ and $(-2,6)$
4. $(-1,-2)$ and $(-2,-1)$

Q. The equation of the directrix of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ is/are

1. $y=\frac{25}{4}$
2. $y=-\frac{25}{4}$
3. $x=-\frac{25}{4}$
4. $x=\frac{25}{4}$

Q. The latus rectum subtends a right angle at the centre of the ellipse, then its eccentricity is

1. $2\sin {18}^{\circ }$
2. $2\cos {18}^{\circ }$
3. $2\sin {54}^{\circ }$
4. $2\cos {54}^{\circ }$

Q. Let the normals at the four points $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ and $(x_4,y_4)$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ be concurrent at some point (called as conormal point). Then $(x_1+x_2+x_3+x_4)(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4})$ is equal to

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

Q. A set of lines $x+y-2+\lambda (2x+y-3)=0$ represents incident rays on an ellipse $S=0$ and
$2x+3y-23+\mu (2x-y-3)=0$ represents the set of reflected rays from the ellipse where $\lambda ,\mu \in R.$ If $P(3,7)$ is a point on the ellipse normal at which meets the major axis at $N$ then

1. Eccentricity of ellipse is $\frac{\sqrt{5}}{2\sqrt{2}+1}$
2. $N$ divides line segment joining two foci in the ratio $2\sqrt{2}:1$
3. Area of triangle formed by point $P$ and two foci is $5$ Sq. unit.
4. Eccentricity of ellipse is $\frac{\sqrt{5}}{2\sqrt{2}-1}$

Q. The radius of the circle passing through the point of intersection of ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $x^2-y^2=0$

1. $\frac{\sqrt{2}ab}{\sqrt{a^2+b^2}}$ unit
2. $\frac{a^2+b^2}{\sqrt{a^2-b^2}}$ unit
3. $\frac{a^2-b^2}{\sqrt{a^2+b^2}}$ unit
4. $\frac{ab}{\sqrt{a^2+b^2}}$ unit

Q.

If $\lambda (3i+2j-6k)$ is a unit vector, then the values of $\lambda$ are:

1. $\pm \frac{1}{7}$

2. $\pm 7$

3. $\pm \sqrt{43}$

4. $\pm \frac{1}{\sqrt{43}}$

5. $\pm \frac{1}{\sqrt{7}}$

Q.

Let the ellipse $C_1:\frac{x^2}{a_1^2}+\frac{y^2}{b_1^2}=1(a_1>b_1)$ and the hyperbola $C_2:\frac{x^2}{a_2^2}-\frac{y^2}{b_2^2}=1$ have the same focus point $F_1$ and $F_2$. If point $P$ is the intersection point of $C_1$​ and $C_2$​ in the first quadrant and $|F_1{F}_2|=2|P{F}_2|$, then which of the following is (are) CORRECT?
( $e_1$ and $e_2$ are eccentricities of ellipse and hyperbola respectively.)

1. $e_1\in (\frac{1}{2},1)$
2. $e_2-e_1\in (\frac{1}{2},\infty )$
3. $e_1\in (\frac{1}{4},\frac{1}{2})$
4. $e_2+e_1\in (\frac{3}{2},\infty )$

Q.

The equation of the ellipse whose foci are ( ±5 , 0 ) and one of its dierctrix is 5x = 36 , is

1. None of these

2. 

3. 

4. 

Q. For the ellipse $x^2+3y^2+2x-12y+10=0,$ which of the following option(s) is/are correct

1. centre is $(-1,2)$
2. length of the axes are $2\sqrt{3}$ and $2$ units
3. eccentricity is $\sqrt{\frac{2}{3}}$
4. length of latus ractum is $\frac{2}{\sqrt{3}}$ units

Q.

If $(-7-24i{)}^{1/2}$=x-iy, then $x^2$+$y^2$=

1. 25

2. -25

3. None of these

4. 15

Q. Define the collections $\{E_1,E_2,E_3,...\}$ of ellipses and $\{R_1,R_2,R_3,...\}$ of rectangles as follows:
$E_1:\frac{x^2}{9}+\frac{y^2}{4}=1;$
$R_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_1;$
$E_n:$ ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1$ of largest area inscribed in $R_{n-1},n>1;$
$R_n:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_n,n>1;$
Then which of the following option is/are correct?

1. The eccentricities of $E_{18}$ and $E_{19}$ are NOT equal
2. $\sum _{n=1}^N\left(areaof{R}_n\right)<24,$for each positive integers $N$
3. The length of latus rectum of $E_9$ is $\frac{1}{6}$
4. The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$

Q. Let $f:R\to R$ be a continuous function satisfying
$f\left(x\right)+\underset{0}{\overset{x}{\int }}tf\left(t\right)dt+x^2=0,$ for all $x\in R.$ Then

1. $\underset{x\to \infty }{lim}f\left(x\right)=2$
2. $\underset{x\to \infty }{lim}f\left(x\right)=-2$
3. $f\left(x\right)$ has more than one point in common with the $x-$axis
4. $f\left(x\right)$ is an odd function

Q. Let $S_1,S_2$ are foci of an ellipse, whose major axis length is $15$ units and $P$ be any point on the ellipse such that perimeter of triangle $P{S}_1{S}_2$ is $20$ units. If $e$ is the eccentricity of the ellipse, then $3e=$

Q. If the extremities of the latus rectum of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ is $(\alpha ,\beta )$, then the distance between the point $P(1,1)$ and $(\alpha ,\beta )$, when $\alpha >0$ is/are

1. $\frac{\sqrt{221}}{25}\text{units}$
2. $\frac{\sqrt{221}}{5}\text{units}$
3. $\frac{\sqrt{541}}{5}\text{units}$
4. $\frac{\sqrt{541}}{25}\text{units}$

Q. If the minor axis of an ellipse subtends an angle of ${60}^{\circ }$ at each focus of the ellipse, then $4e^2=$
(where $e$ is eccentricity)

Q. If the least distance between a point on $x^2+2y^2=6$ and $x+y-7=0$ is $\frac{k}{\sqrt{2}}$ unit, then $k=$

Q. The equation of the axes of the ellipse $4x^2+3y^2-8x+6y-5=0$, are

1. major axis is $y+1=0$
2. minor axis is $y+1=0$
3. minor axis is $x=1$
4. major axis is $x=1$

Q. The relation is a subset of on the power set $P\left(A\right)$ of a set $A$ is ?

1. Symmetric
2. Anti-symmetric
3. Equivalence relation
4. Transitive

Q. A rectangle $ABCD$ has area $200.$Sq.units and an ellipse with area $200\pi$ Sq.units having foci at $B$ and $D$ passes through $A$ and $C$ . If the perimeter of the rectangle is $P$ units then the value of $\frac{P}{20}$ is

Q. Give an example of a relation which is symmetric and transitive but not reflexive

Q. If the points of intersection of the circle $x^2+y^2=16$ with $x-\text{axis}$ are foci of an ellipse and points of intersection of circle $x^2+y^2=16$ with $y-\text{axis}$ are end points of minor axis of the same ellipse, then eccentricity of the ellipse is

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

Q. An ellipse having co-ordinate axes as its axes having lengths $2a$ and $2b$ units respectively, Where $a$ and $b$ are middle terms of a series $a_1,a_2,a_3\cdots a_{10},$ $a_i{a}_{11-i}=5\sqrt{3}\forall i\in N,i<11.$ If $B,F,F^{\prime }$ are one end of minor axis and foci of the ellipse respectively, such that triangle $FB{F}^{\prime }$ is an equilateral triangle, then equation of ellipse is

1. $\frac{x^2}{10}+\frac{2y^2}{15}=1$
2. $\frac{x^2}{15}+\frac{y^2}{5}=1$
3. $\frac{x^2}{10}+\frac{y^2}{15}=1$
4. $\frac{2x^2}{15}+\frac{y^2}{10}=1$