Double Ordinate of Ellipse

Q. If a tangent to the ellipse $x^2+4y^2=4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $BC$ as diameter passes through the point

1. $(-1,1)$
2. $(\sqrt{3},0)$
3. $(1,1)$
4. $(\sqrt{2},0)$

Q. The area of the region bounded by $y-x=2$ and $x^2=y$ is equal to

1. $\frac{2}{3}$
2. $\frac{4}{3}$
3. $\frac{16}{3}$
4. $\frac{9}{2}$

Q. The area bounded by the curve $y=x^3$, x-axis and two ordinates x=1 to x=2 equal to

1. $\frac{15}{2}sq.unit.$
2. $\frac{15}{4}sq.unit.$
3. $\frac{17}{2}sq.unit.$
4. $\frac{17}{4}sq.unit.$

Q.

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

1. $135$

2. $116$

3. $126$

4. $145$

Q.

What are the three dimensions?

Q.

The parametric representation of a point on the ellipse whose foci are $\left(-1,0\right)$ and $\left(7,0\right)$ and eccentricity $\frac{1}{2}$ is

1. $\left(3+8\cos \theta ,4\sqrt{3}\sin \theta \right)$

2. $\left(8\cos \theta ,4\sqrt{3}\sin \theta \right)$

3. $\left(3+4\sqrt{3}\sin \theta ,8\sin \theta \right)$

4. None of the above

Q. The area included by the curve $y=\ln x,x-$axis and the two ordinates at $x=\frac{1}{e}$ and $x=e$ is

1. $2-\frac{2}{e}$
2. $1-\frac{1}{e}$
3. $\frac{2}{e}$
4. $\frac{1}{e}$

Q.

For the ellipse $25x^2+9y^2-150x-90y+225=0$, the eccentricity is equal to

1. $\frac{2}{5}$

2. $\frac{3}{5}$

3. $\sqrt{\frac{15}{24}}$

4. $\frac{4}{5}$

Q. If the point $P$ on the curve, $4x^2+5y^2=20$ is farthest from the point $Q(0,-4)$, then $P{Q}^2$ is equal to:

1. $48$
2. $29$
3. $21$
4. $36$

Q. If the double ordinate of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ passes through the $(5,2),$ then the equation of double ordinate is

1. $y=2$
2. $x=5$
3. $x=6$
4. $y=4$

Q.

What is the eccentricity of an ellipse?

Q. Eccentricity of ellipse $\frac{x^2}{a^2+1}+\frac{y^2}{a^2+2}=1is\frac{1}{\sqrt{3}}$ then length of Latus rectum is

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

Q. The equation of the latus rectum of the ellipse $9x^2+4y^2-18x-8y-23=0$ are

1. $y=\pm \sqrt{5}$
2. $y=-\sqrt{5}$
3. $y=1\pm \sqrt{5}$
4. $y=-1\pm \sqrt{5}$

Q. If the latus rectum of an ellipse $x^2{\tan }^2\phi +y^2{\sec }^2\phi =$ $1$ is $1/2,$ then $\phi$ is

1. $\pi /6$
2. $\pi /2$
3. $\pi /3$
4. $5$ $\pi /12$

Q. If ${\left(\frac{1+i}{1-i}\right)}^3-{\left(\frac{1-i}{1+i}\right)}^3=x+iy,\text{find}(x,y).$

Q. The equation of the latusrecta of the ellipse $9x^2+4^2-18x-8y-23=0$ are

1. $x=1\pm \sqrt{5}$
2. $y=\pm \sqrt{5}$
3. $x=\pm \sqrt{5}$
4. $y=1\pm \sqrt{5}$

Q.

If the latus-rectum of an ellipse is one half of its minor axis, then its eccentricity is

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

Q. Find the eccentricity, foci and the length of the latus-rectum of the ellipse $x^2+4y^2+8y-2x+1=0$.

Q. If the eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is 10, then find the latus rectum of the ellipse.

Q. If the normal at the end of latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through $(0,-b)$, then $e^4+e^2$ (where $E$ is eccentricity) equals

1. square of the eccentricity is equal to the ratio of the minor and major axes
2. ratio of the minor and major axes is $\frac{\sqrt{5}+1}{2}$
3. eccentricity of the ellipse is $\sqrt{\frac{\sqrt{5}-1}{2}}$
4. $noneofthese$

Q. Find the value of $a$ for which the ellipse $\frac{x^2}{y^2}+\frac{y^2}{b^2}=1,(a>b)$, if the extremities of the latus rectum of the ellipse having positive ordinates lie on the parabola $x^2=-2(y-2)$.

Q. If the length of the latusrectum of the ellipse $x^2{\tan }^2\alpha +y^2{\sec }^2\alpha =1$ is $\frac{1}{2}$, then $\alpha$ =

1. 
2. 
3. none
4. 

Q. If the latus rectum of an ellipse is equal to half of minor axis, find its eccentricity.

Q. $AB$ is a diameter of $x^2+9y^2=25$. The eccentric angle of $A$ is $\frac{\pi }{6}$. Then the eccentric angle of $B$ is

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

Q. Locus of the point which divided the double ordinates of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in the ratio $1:2$ internally is

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

Q. Statement-1 : Eccentricity of ellipse whose length of latus rectum is same as distance between is $2sin{18}^o$
Statement-2 : For $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, eccentricity $e=\sqrt{1-\frac{b^2}{a^2}}$

1. STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is correct explanation for STATEMENT-1
2. STATEMENT-1 is true, STATEMENT-2 is true and STATEMENT-2 is not correct explanation for STATEMENT-1
3. STATEMENT-1 is true, STATEMENT-2 is false
4. STATEMENT-1 is false, STATEMENT-2 is true
5. Both STATEMENTS are false

Q. The eccentricity of an ellipse is $\frac{\sqrt{3}}{2}$ its length of latus reetum is

1. $\frac{1}{2}$ (length of major axis)
2. $\frac{1}{4}$ (length of major axis)
3. $\frac{2}{3}$ (length of major axis)
4. $\frac{1}{3}$ (length of major axis)

Q. If the eccentricity of an ellipse is $\frac{5}{8}$ and the distance between its foci is $10$, then find latus rectum of the ellipse.

Q. The length of latus rectum of the ellipse $4x^2+9y^2=36$ is

1. $\frac{4}{3}$
2. $\frac{8}{3}$
3. $6$
4. $12$

Q. Find the eccentricity of that ellipse, whose latus rectum is half of the minor axis.