Standard Form of an Ellipse

Q. The equation of the ellipse with its focus at (6, 2), centre at (1, 2) and which passes through the point (4, 6) is

1. 
2. 
3. 
4. 

Q.

The latusrectum of the hyperbola $\frac{x^2}{4}-\frac{y^2}{12}=1$ is 4 $\frac{1}{2}$ . Its eccentricity e =

1. 4/5

2. 5/4

3. 3/4

4. 4/3

Q.

What is the point of contact between the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and
the tangent $y=mx\pm \sqrt{a^2{m}^2-b^2}$.

1. $[\pm \frac{a^{2}m}{\sqrt{a^2{m}^2-b^2}},\pm \frac{b^2}{\sqrt{a^2{m}^2-b^2}}]$

2. $[\pm \frac{b^2}{\sqrt{a^2{m}^2-b^2}},\pm \frac{a^{2}m}{\sqrt{a^2{m}^2-b^2}}]$

3. $[\pm \sqrt{a^2{m}^2-b^2},\pm \sqrt{a^2{m}^2+b^2}]$

4. $[\pm a^{2}m,\pm b^2]$

Q.

The ends of a rod of length l move on two mutually perpendicular lines. The locus of the point on the rod

which divides it in the ratio 1 : 2 is

1. 

2. 

3. 

4. None of these

Q.

e and $e^1$ are the eccentricities of the hyperbolas $16x^2-9y^2=144$ and $9x^2-16y^2$= - 144 then e - $e^1$ =

1. 2

2. 3/2

3. 1

4. 0

Q. P is a variable point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with AA’ as the major axis. Then the maximum value of the area of $△AP{A}^{\prime }$ is

1. 
2. none
3. 2ab
4. ab

Q.

For each point (x, y) on an ellipse, the sum of the distances from (x, y) to the points (2, 0) and (-2, 0) is 8. Then the positive value of x so that (x, 3) lies on the ellipse is ___

1. 2

2. 4

3. $2\sqrt{3}$

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

Q.

Find the coordinatesof the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{36}+\frac{y^2}{16}=1.$

Q.

Find the equation of the ellipse whose foci are at $(0,\pm 4)$ and $e=\frac{4}{5}.$

Q.

The foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}+\frac{y^2}{25}=\frac{1}{13}$ are the same.

The value of $b^2$ is ___ .

Q.

If the semi-major axis of an ellipse is 3 and the latus rectum is 16/9, then the standard equation of the ellipse is

1. $\frac{x^2}{9}+\frac{y^2}{8}=1$

2. $\frac{x^2}{9}+\frac{3y^2}{8}=1$

3. $\frac{x^2}{9}+\frac{8y^2}{3}=1$

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

Q.

A parabolic reflector is 9 cm deep and its diameter is 24 cm. How far is its focus from the vertex?

Q.

The sides of a rectangle are $x=0,y=0,x=4,y=3$. The equation of the straight line having slope $\frac{1}{2}$ that divides the rectangle into two equal halves, is

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

Q.

Let S and T be the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{8}=1$. If P(x, y) is any point on the ellipse, then the maximum area of the triangle PST (in square units) is ___

1. $2\sqrt{2}$

2. $4\sqrt{2}$

3. $8\sqrt{2}$

4. 8

Q.

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

1. 

2. 

3. 

4. 

Q.

An ellipse with centre at (0, 0) cuts x axis at (3, 0) and (-3, 0). If its e=$\frac{1}{2}$ then the length of the Semiminor axis is

1. 

2. 

3. 

4. 3√3/2

Q. The function $t$ which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t\left(C\right)=\frac{9C}{5}+{32}^o$
Find :

Q. In an ellipse the distance between the foci is 6 and it’s minor axis is 8. Then its eccentricity is

1. 
2. 
3. 
4. 

Q.

If p is the length of perpendicular from the origin to the line whose intercepts on the axes are a and b then show that $\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}$

Q. Find equation of the line passing through the point $(2,2)$ and cutting off intercepts on the axes whose sum is $9$.

Q. If $e_{1}and{e}_2$ are the eccentricities of a hyperbola and its conjugate respectively, Then $e_1^{-2}+e_2^{-2}=1$.

1. False

2. True

Q. An ellipse has a focus at $(ae,0)$ and directrix along $x=\frac{a}{e}$. If $b^2=a^2(1-e^2)$, then the equation of ellipse is

1. x2a2+y2b2=1
2. x2a2-y2b2=1
3. x2a2+y2b2=0
4. x2a2+y2b2=-1

Q. Find the real numbers $x$ and $y$ , if $(x-iy)(3+5i)$ is the conjugate of $-6-24i$.

Q. The equation, $\frac{x^2}{10-a}+\frac{y^2}{4-a}=1$ represents an ellipse if

1. $a>4$
2. $a<4$
3. $4<a<10$
4. $a>10$

Q.

If the distance between a focus and corresponding directrix of an ellipse be 8 and the eccentricity be $\frac{1}{2}$ ,

then length of the minor axis is

1. 3

2. 

3. 6

4. None of these

Q. Show that $\frac{1\times 2^2+2\times 3^2+...+n\times {(n+1)}^2}{1^2\times 2+2^2\times 3+...+n^2\times (n+1)}=\frac{3n+5}{3n+1}$

Q. Find the equation of the hyperbola satisfying the given conditions: Foci $(\pm 5,0)$ the
transverse axis is of length $8$

Q. Find the equation of the right bisector of the line segment joining the points $(3,4)$ and $(-1,2)$.

Q. A circle of radius ‘r’ is concentric with the Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ Then inclination of common tangent with major axis is _______(b<r<a)

1. 
2. 
3. 
4. 

Q. Find the foot of the perpendicular of (3, 6) on the line x - 2y + 4 = 0

1. (2, 3)

2. (5, 2)

3. (4, 4)

4. (-2, 1)