Terminologies Related to an Ellipse

Q.

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $e$ of the ellipse satisfies

1. $e^4+2e^2-1=0$

2. $e^2+2e-1=0$

3. $e^4+e^2-1=0$

4. $e^2+e-1=0$

Q.

The eccentricity of an ellipse is $\frac{2}{3}$ , latus rectum is 5 and centre is (0, 0). The equation of the ellipse is

1. 

2. 

3. 

4. 

Q.

In an Ellipse distance between the foci is 6 and the length of minor axis is 8. Its eccentricity is

1. 2/5

2. 4/5

3. 3/5

4. 1/3

Q.

If the eccentricity of an ellipse be $\frac{1}{\sqrt{2}}$ , then its latus rectum is equal to its

1. Major axis

2. Semi-major axis

3. Semi-minor axis

4. Minor axis

Q.

The latus rectum of an ellipse is 10 and the minor axis is equal to the distance between the foci. The equation of the ellipse is

1. 

2. 

3. 

4. None of these

Q.

The points (-a, -b), (0, 0), (a, b) and ($a^2$, ab) are

1. Collinear

2. Vertices of a rectangle

3. Vertices of a parallelogram

4. None of these

Q.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.

$\frac{x^2}{49}+\frac{y^2}{36}=1$

Q.

In a hyperbola the latusrectum equals to semitransverse axis, then its eccentricity is

1. 3/2

2. 4/3

3. 5/2

4. 

Q. Two rods of lengths a and b slides along the x-axis and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is

1. 
2. 
3. 
4. 

Q. A circle cuts the X-axis and Y-axis such that intercept on X-axis is a constant a and intercept on Y-axis is a constant b. Then eccentricity of locus of centre of circle is

1. 1
2. 
3. 
4. 

Q.

The eccentricity of the ellipse $12x^2+7y^2=84$ is equal to

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

2. $\sqrt{\frac{5}{12}}$

3. $\frac{\sqrt{5}}{12}$

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

Q.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{100}+\frac{y^2}{400}=1.$

Q.

The foci of the conic section $25x^2+16y^2-150x=175$ are

1. $(0,\pm 3)$

2. $(0,\pm 2)$

3. $(3,\pm 3)$

4. $(0,\pm 1)$

Q. The eccentricity of an ellipse whose minor axis is double the latus rectum is

1. 34
2. 32
3. 13
4. 123

Q.

Given standard equation of ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, with eccentricity e.

Match the following
$\begin{array}{cc}\text{a) Major axis}& \text{i) 2a(1}-{\text{e}}^2)\\ \text{b) Minor axis}& \text{ii) y=0}\\ \text{c) Double ordinate}& \text{iii) x=0}\\ \text{d) Latus Rectum length}& \text{iv) x=}\frac{\text{-a}}{\text{e}}\\ & \text{v)}\sqrt{1-\frac{{\text{b}}^{\text{2}}}{{\text{a}}^{\text{2}}}}\end{array}$

1. a = ii, b = iii, c = iii, d = i
2. a = ii, b = iv, c = iii, d = v
3. a = iii, b = ii, c = vi, d = i
4. a = v, b = ii, c = vi, d = i

Q.

A point $P$ on an ellipse is at a distance of $6$ units from a focus. The eccentricity of the ellipse is $\frac{3}{5}$, then the distance of $P$ from the corresponding directrix is

1. $\frac{8}{5}$

2. $\frac{5}{8}$

3. $10$

4. $12$

5. None of these

Q.

Find the lengths of the axes; the coordinates of the vertices and the foci the eccentricity and length of the latus rectum of the hyperbola $y^2-16x^2=16.$

Q.

Match the following for ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$ with eccentricity $e$.

1. $(\pm ae,0)$
2. $x=\pm \frac{a}{e}$
3. $(\pm a,0)$
4. 1-b2a2

Q. If the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ meet the ellipse $\frac{x^2}{1}+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-10b+25$, then the value b does not satisfy

1. 
2. (4, 6)
3. [4, 6]
4. 

Q.

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{4}+\frac{y^2}{25}=1.$

Q. Match the following for ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, with eccentricity $e.$

1. $2a(1-e^2)$
2. $y=0$
3. $x=0$
4. $x=2a$

Q. The slope of a line is double of the slope of another line. If tangent of the angle between them is $\frac{1}{3}$, find the slopes of the lines.

Q.

If the major axis is "n” times the minor axis of the ellipse, then eccentricity is

1. 

2. 

3. 

4. 

Q.

If $z_1,z_2,z_3,z_4$ are the four complex numbers represented by the vertices of a quadrilateral taken in order such that $z_1-z_2=z_3-z_4$ and amp $\frac{z_4-z_1}{z_2-z_1}=\frac{\pi }{2}$, then the quadrilateral is a

1. square
2. rhombus
3. rectangle
4. a cyclic quadrilateral

Q.

Given standard equation of ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, with eccentricity e

Match the following
$\begin{array}{cc}\text{a)Focus}& \text{i) (ae, 0)}\\ \text{b)Directrix}& \text{ii) (a, 0)}\\ \text{c)Eccentricity}& \text{iii) x=}\frac{\text{a}}{\text{e}}\\ \text{d)Vertices}& \text{iv) (-ae, 0)}\\ & \text{v) x=}\frac{\text{-a}}{\text{e}}\\ & \text{vi)}\sqrt{\text{1-}\frac{{\text{b}}^{\text{2}}}{{\text{a}}^{\text{2}}}}\end{array}$

1. a = i, iv, b = iii, v , c =vi, d = ii
2. a = i, b = iii, v , c =vi, d = ii
3. a = i, b = ii, v , c =vi, d =iii
4. a = i, iv, b = ii, v , c =vi, d =iii

Q.

If a point P(x, y) moves along the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if C is the centre of the ellipse, then, 4 max{CP} + 5 min{CP}=___ .

1. 30

2. 40

3. 45

4. 25

Q.

In a hyperbola e=2 and the length of semitransverse axis is 3 and the length of conjugate axis is

1. 

2. 

3. 

4. 

Q.

A point P on the ellipse is at a distance of 6 units from the focus. If the eccentricity of the ellipse is 0.6, then the distance of P from the corresponding directrix is ___

Q. Draw the graph of $y=\left[e^{\left[x\right]}\right]$

1. 

2. 

3. 

4. 

Q.

How many of the following statements are correct?

(a) The focus of $x^2=4ay$ is (0, a)

(b) The directrix of $y^2=-4ax$ is x + a = 0

(c) The end points of latus rectum of $x^2=-4ay$ is (a, 2a) and (a, -2a)

(d) The parabolas $x^2=4ay$ and $y^2=4ax$ are equal.

__