Eccentricity of Ellipse

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

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

Q. The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is

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

Q. The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is

1. $\frac{\sqrt{5}+1}{2}$
   
2. none of these
3. $\frac{\sqrt{5}-1}{2}$
4. $\frac{\sqrt{5}-1}{4}$
   

Q. If the major axis of an ellipse is three times the minor axis, then its eccentricity is equal to

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

Q. If a variable point $P$ on an ellipse with eccentricity $e$ is joined to it's focii $S,S^{\prime }$.Then locus of incentre of $△PS{S}^{\prime }$ is another ellipse whose eccentricity is

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

Q. The eccentricity of the conic $9x^2+25y^2=225$ is

1. 2/5
   
2. 4/5
   
3. 1/5
4. 1/3
   

Q. Let the length of the latus rectum of an ellipse with its major-axis along $x$-axis and centre at the origin, be $8$. If the distance between the foci of this ellipse is equal to the length of its minor axis, then which one of the following points lies on it?

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

Q. Let $S(3,4)$ and $S^{\prime }(9,12)$ be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus $S$ to a tangent of the ellipse is $(1,-4),$ then the eccentricity of the ellipse is

1. $\frac{4}{5}$
2. $\frac{5}{7}$
3. $\frac{7}{13}$
4. $\frac{5}{13}$

Q. The equation of the ellipse, whose axes are of lengths $6$ and $2\sqrt{6}$ and their equations are $x-3y+3=0$ and $3x+y-1=0$ respectively, is

1. $29x^2-6xy+21y^2+6x-58y-151=0$
2. $21x^2-6xy+29y^2+58x-6y-151=0$
3. $29x^2-6xy+21y^2+6x-58y+151=0$
4. $21x^2-6xy+29y^2+6x-58y-151=0$

Q. The eccentricity of the ellipse $25x^2+16y^2=400$ is

1. 3/5
   
2. 1/3
   
   
3. 2/5
   
   
4. 1/5

Q. Find the eccentricity of the hyperbola $9y^2-4x^2=36$

Q. Find the eccentricity equation of an ellipse whose latus-rectum is
(i) half of its minor axis
(ii) half of its major axis.

Q. Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

Q.

The eccentricity of the ellipse which meets the straight line $\frac{x}{7}+\frac{y}{2}=1$ on the axis of x and the straight line $\frac{x}{3}-\frac{y}{5}=1$ on the axis of y and whose axes lie along the axes of coordinates, is

1. none of these

2. $\frac{3\sqrt{2}}{7}$

3. $\frac{2\sqrt{3}}{7}$

4. $\frac{\sqrt{3}}{7}$

Q.

If the length of the major axis of an ellipse is $\frac{17}{8}$ times the length of the minor axis, then the eccentricity of the ellipse is

1. $\frac{8}{17}$

2. $\frac{15}{17}$

3. $\frac{9}{17}$

4. $\frac{2\sqrt{2}}{17}$

5. $\frac{13}{17}$

Q. If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is inscribed in a rectangle whose length to breadth ratio is $2:1$ in such a manner that it touches the sides of rectangle, then the area of the rectangle is

1. $4\left(\frac{a^2+b^2}{7}\right)$
2. $4\left(\frac{a^2+b^2}{3}\right)$
3. $12\left(\frac{a^2+b^2}{5}\right)$
4. $8\left(\frac{a^2+b^2}{5}\right)$

Q. A point moves so that its distance from the point $(2,0)$ is always $\frac{1}{3}$ of its distance from the line $x-18=0.$ If the locus of the point is a conic, then

1. Locus of the point will be $\frac{x^2}{36}+\frac{y^2}{32}=1.$
2. Locus of the point will be $\frac{x^2}{18}+\frac{y^2}{16}=1.$
3. Length of the latus rectum of the conic $=\frac{32}{3}$ units
4. Length of the latus rectum of the conic $=\frac{16\sqrt{2}}{3}$ units

Q. The eccentricity of the ellipse $4x^2+9y^2+8x+36y+4=0$ is

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

Q. If the length of latus rectum of a horizontal ellipse $x^2{\tan }^2\alpha +y^2{\sec }^2\alpha =1(\alpha \ne \frac{\pi }{2})$ is $\frac{1}{2}$, then the possible value(s) of $\alpha$ in $(0,\pi )$ is/are

1. $\frac{\pi }{12}$
2. $\frac{\pi }{6}$
3. $\frac{5\pi }{12}$
4. $\frac{7\pi }{12}$

Q.

Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
(i)$4x^2+9y^2=1$
(ii)$5x^2+4y^2=1$
(iii)$4x^2+3y^2=1$
(iv)$25x^2+16y^2=1600$
(v)$9x^2+25y^2=225$

Q.

An ellipse is described by using an endless string which is passed over two pins.

If the axes are $6cm$ and $4cm$, the length of the string and the distance between the pins are ____________

Q. The eccentricity of the ellipse represented by the equation $25x^2+16y^2-150x-175=0$ is

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

Q. The eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ if its latus-rectum is equal to one half of its minor axis, is

1. $\frac{\sqrt{3}}{2}$
   
2. $\frac{1}{2}$
   
3. $\frac{1}{\sqrt{2}}$
   
4. none of these

Q. Let $AB$ be a line segment of length $4$ units with $A$ on line $y=2x$ and $B$ on the line $y=x,$ the locus of middle points of all such line segments is a

1. Parabola
2. Ellipse
3. pair of straight lines
4. circle

Q.

Find the lengths of transverse axis and conjugate axis, eccentricity, the co-ordinates of foci, vertices, length of the latus-rectum, and equations of the directrices of the following hyperbola $16x^2-9y^2=-144$.

Q.

If for the ellipse $\left(\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}\right)+\left(\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}\right)=1$, the y-axis is the minor axis and the length of the latus rectum is one half of the length of its minor axis, then its eccentricity is

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

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

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

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

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

Q. The points of intersection of the two ellipse $x^2+2y^2-6x-12y+23=0$ and $4x^2+2y^2-20x-12y+35=0$

1. lie on a circle centered at $(\frac{-8}{3},-3)$ and of radius $\frac{1}{3}\sqrt{\frac{47}{2}}$ unit
2. lie on a circle centered at $(\frac{8}{3},3)$ and of radius $\frac{1}{3}\sqrt{\frac{47}{2}}$ unit
3. lie on a circle centered at $(8,9)$ and of radius $\frac{1}{3}\sqrt{\frac{47}{2}}$ unit
4. are not concyclic

Q. The equation of the ellipse which passes through origin and has its foci at the points $(1,0)$ and $(3,0),$ is

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

Q. Write the eccentricity of the ellipse $9x^2+5y^2-18x-2y-16=0$

Q. If S and S are two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and B is an end of the minor axis such that $△BS{S}^{\prime }$ is equilateral, then write the eccentricity of the ellipse.