Eccentricity of Ellipse

Q.

Find the equation of an ellipse whose major axis lies on the x-axis and which passes through the points (4, 3) and (6, 2).

Q. 42. The distance between the foci of an ellipse is 10 and length of latus rectum is 15; find its equation referred to its axes as axes of co-ordinates

Q.

The equation of the two tangents from $\left(-5,-4\right)$ to the circle $x^2+y^2+4x+6y+8=0$ are

1. $x+2y+13=0,2x-y+6=0$

2. $2x+y+13=0,x-2y=6$

3. $3x+2y+23=0,2x-3y+4=0$

4. $x-7y=23,6x+13y=4$

Q.

Find the eccentricity of ellipse whose minor axis is double the latus rectum

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

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

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

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

Q.

Tangents are drawn from the point $\left(17,7\right)$ to the circle $x^2+y^2=169$

Statement I: The tangents are mutually perpendicular.

Statement II: The locus of the points from which mutually perpendicular tangents can be drawn to the given circles is $x^2+y^2=338$

1. Statement I is correct, Statement II is correct; Statement II is the correct explanation for Statement I

2. Statement I is correct, Statement II is correct; Statement II is not a correct explanation for Statement I

3. Statement I is correct, Statement II is correct

4. Statement I is incorrect, Statement II is correct

Q. 7. the extreme points of the latus rectum of a parabola are (7, 5) and (7, 3) . Find the equation of a parabola.

Q. Find the length of the latus rectum of the ellipse whose foci are (-2, -1) and (1, 2) and one of the directrices is x+y=5.

Q. The abscissa of a point on the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at a distance of $\frac{5}{4}$ unit from focus is

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

Q. The equation(s) of standard ellipse which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$, is/are

1. $3x^2+5y^2=32$
2. $3x^2+5y^2=48$
3. $5x^2+3y^2=32$
4. $5x^2+3y^2=48$

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. 14. Find the length of the latus rectum of the ellipse whose foci are (-2, -1) and (1, 2) and one of the directrices is x+y = 5.

Q. If $e_1$ and $e_2$ are the eccentricities of the ellipse , $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on th ellipse , $15x^2+3y^2=k$. Then $k$ is equal to :

1. $14$
2. $15$
3. $17$
4. $16$

Q.

An ellipse has OB as a semi-minor axis. F and F' are its foci and the angle FBF' is a right angle. Then, the eccentricity of the ellipse is ___

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

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

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

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

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. $\frac{5}{13}$
4. $\frac{7}{13}$

Q. Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
(i) 4x² + 9y² = 1
(ii) 5x² + 4y² = 1
(iii) 4x² + 3y² = 1
(iv) 25x² + 16y² = 1600.

Q. find eccentricity of an ellipse if its latus rectum is 1/3rd of its minor axis

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 equation of the ellipse centred at $(1,2)$, one focus at $(6,2)$ and passing through the point $(4,6),$ is

1. $\frac{(x-1{)}^2}{45}+\frac{(y-2{)}^2}{20}=1$
2. $\frac{(x-1{)}^2}{5}+\frac{(y-2{)}^2}{20}=1$
3. $\frac{(x-1{)}^2}{20}+\frac{(y-2{)}^2}{45}=1$
4. $\frac{(x-2{)}^2}{20}+\frac{(y-1{)}^2}{45}=1$

Q. If sum of distance of a point from two perpendicular lines in a plane is $1$, then its locus is ?

1. Square
2. A straight line
3. Circle
4. An intersecting line

Q.

The eccentricity of the ellipse $a{x}^2+b{y}^2+2fx+2gy+c=0$ if axis of ellipse parallel to x-axis is

1. $\left(\sqrt{\frac{b-a}{b}}\right)$

2. $\left(\sqrt{\frac{a+b}{b}}\right)$

3. $\left(\sqrt{\frac{a+b}{a}}\right)$

4. none of these

Q. The eccentricity of the ellipse, whose end points of major axis and minor axis are $(\pm \sqrt{5},0)$ and $(0,\pm 1)$ respectively, is

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

Q. If tangent to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the axes at $A$ and $B,$ then the locus of mid point of $AB$ is

1. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=2$
2. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=4$
3. $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$
4. none of these

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. If the length of the major axis of a vertical ellipse is three times length of the minor axis, then its eccentricity is equal to

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

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

Q. The difference between the lengths of the major axis and the latus-rectum of an ellipse is
(a) ae
(b) 2ae
(c) ae²
(d) 2ae²

Q. If $e_1$ and $e_2$ are the eccentricities of the ellipse , $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on th ellipse , $15x^2+3y^2=k$. Then $k$ is equal to :

1. $14$
2. $15$
3. $17$
4. $16$

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