Eccentricity

Q.

How do you Convert Cartesian to Vector?

Q.

The eccentricity of the conic $4x^2+16y^2-24x-32y=1$ is:

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

2. $\sqrt{3}$

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

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

Q.

If the eccentricity of the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$and the hyperbola $\frac{x^2}{64}-\frac{y^2}{b^2}=1$are reciprocals of each other, then $b^2=?$

1. $192$

2. $64$

3. $16$

4. $32$

5. $128$

Q. In a triangle ABC, the bisectors of angles B and C lie long the lines $x=y$ and $y=0$. If A is $(1,2)$, then the equation of line BC is

Q. If $e_1$ is the eccentricity of the conic $9x^2+4y^2=36$ and $e_2$ is the eccentricity of the conic $9x^2-4y^2=36$, then

Q. The equation of the ellipse, whose focus is the point $(-1,1)$, whose directrix is the straight line $x-y+3=0$ and whose eccentricity is $\frac{1}{2}$ is :

1. $(x+1{)}^2+(y-1{)}^2=\frac{1}{2}(x-y+3{)}^2$
2. $(x+1{)}^2+(y-1{)}^2=\frac{1}{8}(x-y+3{)}^2$
3. $(x+1{)}^2+(y-1{)}^2=\frac{1}{8}(x-y+1{)}^2$
4. $(x+1{)}^2+(y-1{)}^2=\frac{1}{6}(x-y+3{)}^2$

Q.

The peak value of AC is $2\sqrt{2}\mathrm{Ampere}$. Its apparent value will be?

1. $0$

2. $1\mathrm{Ampere}$

3. $2\mathrm{Ampere}$

4. $3\mathrm{Ampere}$

Q. The equation of a conic with directrix $3x+4y-5=0$ and focus (0, 0) is given as $x^2+y^2=(3x+4y-5{)}^2$. Find eccentricity of the conic.
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Q. Find the equation of the ellipse whose foci are $(4,0)$ and $(-4,0),$ eccentricity $e=\frac{1}{3}$.

Q. If the normal at $(c{t}^1,{\frac{c}{t}}^1)$ on the hyperbola $xy=c^2$ cuts the hyperbola again at $(c{t}^2,{\frac{c}{t}}^2)$, then $t^1{t}^2=$

1. -2
2. -1
3. 2
4. 1

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

Q. If the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be $e$ and $e_1$, then $\frac{1}{e^2}+\frac{1}{e_1^2}=$.

1. $2$
2. $1$
3. None of these
4. $3$

Q. If $e_1$ and $e_2$ are the eccentricities of two conics with $e_1^2+e_2^2=3$, then the conics are.

1. Ellipses
2. Parabolas
3. Circles
4. Hyperbolas

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

Q. If the latus-rectum of an ellipse be equal to half of its minor axis, then its eccentricity is $\underset{–}{}$

Q. Equations $x=a\cos \theta$ and $y=b\sin \theta$ represent a conic section whose eccentricity $e$ is given by

1. $e^2=\frac{a^2+b^2}{a^2}$
2. $e^2=\frac{a^2-b^2}{a^2}$
3. None of these
4. $e^2=\frac{a^2+b^2}{b^2}$

Q. Find the vertex focus, equation of directrix and equation of axis of the $y^2-x+4y+5=0$.

Q. If $\overline{a},\overline{b},\overline{c}$ are three non-zero vectors such that $\overline{a}+\overline{b}+\overline{c}=0$ then the value of $\overline{a}\cdot \overline{b}+\overline{b}\cdot \overline{c}+\overline{c}\cdot \overline{a}$ is

1. less than zero
2. equal to zero
3. 3
4. greater than zero

Q. Eccentricity of the hyperbola whose asymptotes are given by $3x+2y+5=0$ and $2x+3y+5=0$ is

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

Q. The eccentricity of the hyperbola passing through the points $(3,0)$ and $(3\sqrt{2},2)$ is _________

Q. Let circles $C_1$ and $C_2$ an Argand plane be given by $|z+1|=3$ and $|z-2|=7$ respectively. If a variable circle $|z-z_0|=r$ be inside circle $C_2$ such that it touches $C_1$ externally and $C_2$ internally then locus of $z_0$ describes a conic $E$ whose eccentricity is equal to

1. $\frac{5}{10}$
2. $\frac{1}{10}$
3. $\frac{3}{10}$
4. $\frac{7}{10}$

Q. A straight line drawn through the common focus S' of a number of conics meets them in the points $P_1,P_2,....;$ on it is taken a point Q such that the reciprocal of SQ is equal to the sum of the reciprocals of $S{P}_1,S{P}_2,...$ Prove that the locus of Q is a conic section whose focus is O, and show that the reciprocal of its latus rectum is equal to the sum of the reciprocals of the latera recta of the given conics.

Q. The eccentricity of the conic represented by $\sqrt{(x+2{)}^2+y^2}+\sqrt{(x-2{)}^2+y^2}=8$ is

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

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

Q. Find the eccentricity of an ellipse, if its latus rectum be equal to one half its minor axis.