Eccentricity of Ellipse

Q.

The equation of two circles which touch the $y$ - axis at $\left(0,3\right)$ and make an intercept of $8$ units on $x$ - axis , are

1. $x^2+y^2\pm 10x-6y+9=0$

2. $x^2+y^2\pm 6x-10y+9=0$

3. $x^2+y^2-8x\pm 10y+9=0$

4. $x^2+y^2-10x\pm 6y+9=0$

Q.

If the line $y=7x-25$ meets the circle$x^2+y^2=25$ in the points $A,B$, then the distance between $A$ and $B$ is

1. $\sqrt{10}$

2. $10$

3. $5\sqrt{2}$

4. $5$

Q.

The ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ have in common:

1. centre only

2. centre, foci and directrices

3. centre, foci and vertices

4. centre and vertices

Q.

The equation of a sphere having centre $(-1,2,-3)$ and radius $3$ units is

1. $x^2+y^2+z^2–2x–4y–6z=0$

2. $x^2+y^2+z^2+2x–4y+6z+5=0$

3. $x^2+y^2+z^2–2x–4y–6z–5=0$

4. None of these

Q.

If $p\text{and}q$ be the longest distance and the shortest distance respectively of the point $\left(–7,2\right)$ from any point $\left(\alpha ,\beta \right)$ on the curve whose equation is $x^2+y^2–10x–14y–51=0$, then G.M. of $p\text{and}q$ is equal to

1. $2\sqrt{11}$

2. $5\sqrt{5}$

3. $13$

4. None of these

Q. The eccentricity of the ellipse $3x^2+2y^2=6$ is $\sqrt{3}$.

1. False
2. True

Q.

In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, the eccentricity of the ellipse is

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

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

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

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

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

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

Q. Which of following is incorrect regarding eccentricity of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\text{and}{c}^2=a^2+b^2?$

1. $e=\sqrt{1+\frac{a^2}{b^2}}$
2. $e=\sqrt{1+\frac{b^2}{a^2}}$
3. $e=\frac{\text{Distance from centre to focus}}{\text{Distance from centre to Vertex}}$
4. $e=\frac{c}{a}$

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. The eccentricity of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$whose latus rectum is half of its major axis, is .

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

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 length of minor axis (along $y$-axis) of an ellipse of the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line $x+6y=8$, then its eccentricity is:

1. $\frac{1}{2}\sqrt{\frac{5}{3}}$
2. $\frac{1}{2}\sqrt{\frac{11}{3}}$
3. $\sqrt{\frac{5}{6}}$
4. $\frac{1}{3}\sqrt{\frac{11}{3}}$

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. Let $x+y=0$ and $2x-y+3=0$ be the major and minor axis of an ellipse respectively. If the foot of perpendicular drawn from vertex of the parabola $x^2-4x+4y+16=0$ to these lines is focus and one endpoint of minor axis respectively, then the eccentricity of the ellipse is

1. $\frac{7}{\sqrt{59}}$
2. $\frac{7}{8}$
3. $\frac{5}{\sqrt{59}}$
4. $\frac{4}{\sqrt{53}}$

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. The equation(s) of an 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.

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

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

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

4. none of these

Q. Let $A$ and $B$ be two drawing pins such that $AB=10$ units .Consider a string whose end points are at $A,B$ and length is of $16$ units. If the point of pencil moves on paper such that fixed ends of string are always tight the area encolsed by curve traced by pencil is

1. $36\pi$ sq. units
2. $100\pi$ sq. units
3. $16\sqrt{39}\pi$ sq. units
4. $8\sqrt{39}\pi$ sq. units

Q. A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

1. $\frac{x^2}{9}-\frac{y^2}{4}=1$
2. $\frac{x^2}{9}-\frac{y^2}{16}=1$
3. $x^2-y^2=9$
4. $\frac{x^2}{9}-\frac{y^2}{25}=1$

Q. If the major axis of a vertical ellipse is three times 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. The length of minor axis (along $y$-axis) of an ellipse of the standard form is $\frac{4}{\sqrt{3}}$. If this ellipse touches the line $x+6y=8$, then its eccentricity is:

1. $\frac{1}{2}\sqrt{\frac{5}{3}}$
2. $\frac{1}{2}\sqrt{\frac{11}{3}}$
3. $\sqrt{\frac{5}{6}}$
4. $\frac{1}{3}\sqrt{\frac{11}{3}}$

Q. A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

1. $\frac{x^2}{9}-\frac{y^2}{4}=1$
2. $\frac{x^2}{9}-\frac{y^2}{16}=1$
3. $x^2-y^2=9$
4. $\frac{x^2}{9}-\frac{y^2}{25}=1$

Q. Read the following statements carefully and choose the correct option(s)
$\text{(I)}\frac{x^2}{25}+\frac{y^2}{16}=1$ and $12x^2-4y^2=27$ intersect orthogonally.

$\text{(II)}$Locus of vertex of a parabola with focus at $(2,3)$ and length of latus rectum is $8$, is a circle.

$\text{(III)}$The two circles $x^2+y^2-10x+4y-20=0$ and $x^2+y^2+14x-6y+22=0$ intersect each other.

1. $\text{(I)}$ is correct
2. $\text{(II)}$ is correct
3. $\text{(III)}$ is incorrect
4. $\text{(III)}$ is correct

Q. A ray emanating from point $A$ $(-3,0)$ incidents at the point $P$ on the surface of the ellipse $16x^2+25y^2=400$ and gets reflected parallel to minor axis of the ellipse and again reflected at point $Q$ on the surface of the ellipse, then area of $\Delta APQ$ is $\frac{a}{b},$ where $\frac{a}{b}$ is in the smallest form. Then the value of $a+b$ is

Q.

Equation of the ellipse with foci ($\pm ,0$) and e = $\frac{1}{4}$ is

1. $\frac{x^2}{16}+\frac{y^2}{14}$ = 1

2. $\frac{x^2}{18}+\frac{y^2}{16}$ = 1

3. $\frac{x^2}{20}+\frac{y^2}{18}$ = 1

4. $\frac{x^2}{32}+\frac{y^2}{30}$ = 1

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. $5x^2+3y^2=48$