Directrix of Ellipse

Q. Which of the following is/are true about the ellipse $x^2+4y^2-2x-16y+13=0$

1. The length of latus rectum of the ellipse is $1$ unit
2. Distance between foci of the ellipse is $4\sqrt{3}$ unit
3. Sum of the focal distances of a point $P(x,y)$ on the ellipse is $4$ unit
4. If the tangents drawn at the vertices of the ellipse meets line $y=3$ at the points $P$ and $Q$ then $PQ$ always subtends a right angle at any of its foci

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 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. $4\sqrt{2}$

3. 6

4. None of these

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

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

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. $4\sqrt{2}$

3. 6

4. None of these

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

Q. The eccentricity of an ellipse, with its centre at the origin, is $\frac{1}{2}$. If the equation of one of its directrices is x = 4, then the equation of the ellipse is

1. $3x^2+4y^2=1$
2. $4x^2+3y^2=1$
3. $4x^2+3y^2=12$
4. $3x^2+4y^2=12$

Q. A ray eminating from the point $(-3,0)$ is incident on the ellipse $16x^2+25y^2=400$ at the point $P$ with ordinate $4.$ Equation of the reflected ray is

1. $4x+3y+12=0$
2. $4x+3y=12$
3. $3x+4y=16$
4. $3x+4y+9=0$