Conjugate Hyperbola

Q.

Director Circle and directrix are same for a parabola

1. False

2. True

Q. Equation of the chord of the hyperbola $25x^2-16y^2=400$ which is bisected at the point (6, 2), is

1. 75x - 16y = 418
2. 16x – 75y = 418
3. 25x - 4y = 400
4. None of these

Q. From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$ . Then, area cut-off by the chord of contact on the asymptotes is equal to

1. a/2 sq unit
2. ab sq unit
3. 2ab sq unit
4. 4ab sq unit

Q.

What the equation of the auxiliary circle of a hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$

1. 

2. 

3. 

4. 

Q.

Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$49y^2-16x^2=784$

Q. Conjugate of the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is .

1. x2a2+yb2=1
2. x2a2-y2b2=1
3. x2a2-y2b2=-1
4. -x2a2-y2b2=1

Q.

Find the coordinates of the foci, the vertices, the eccentricity and the length of the latus rectum of the hyperbola,

$9y^2-4x^2=36$

Q.

Find the coordinates of the foci, the vertices, the accentricity and the length of the latus rectum of the hyperbola,

$\frac{y^2}{9}-\frac{x^2}{27}=1$

Q.

If the chord joining the points $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$ of the parabola $y^2=4ax$ passes through the focus of the parabola, then

1. 

2. 

3. 

4. 

Q. Which of the following represents the
director circle for the parabola $y^2=4ax?$

1. $x^2+y^2=a^2$
2. $x^2+y^2=1$
3. $x+a=0$

4. $x=a$

Q.

What is the conjugate hyperbola of the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

2. $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

3. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1$

4. $-\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Q.

The locus of the mid-point of the line segment joining the focus ot a moving point on the parabola $y^2$ = 4ax is another parabola with directrix

1. $x=\frac{a}{2}$

2. x = - a

3. x = 0

4. $x=-\frac{a}{2}$

Q.

If the chord of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the middle point of these chord is _____

1. y^2\)

2. y^2\)

3. y\)

4. y^2\)

Q.

Find the equation of the hyperbola satisfying the given conditions,

Vertices $(\pm 2,0),foci(\pm 3,0)$

Q.

The distance between the foci of the hyperbola $9x^2-16y^2+18x+32y-151$ = 0 is

1. 2

2. 8

3. 10

4. 6

Q. If $S_1$ and $S_2$ are the foci of the hyperbola whose transverse axis length is $4$ units and conjugate axis length is $6$ units, $S_3$ and $S_4$ are the foci of the conjugate hyperbola, then the area of the quadrilateral $S_1{S}_3{S}_2{S}_4$ is sq. units

Q. if e₁ be the eccentricity of the ellipse x²/16+y²/25=1 and e₂ is the eccentricity of the hyperbola passing through the foci of the ellipse and e₁e₂=1 then equation of hyperbola is

Q.

A hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its conjugate hyperbola. The foci points of both hyperbolas are connected as shown. Then $S_1{S}_3{S}_2{S}_4$ always forms a

1. Rhombus

2. Trapezium

3. Square

4. None of the above

Q. The equation of the conjugate hyperbola of the hyperbola $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-3=0$ is

1. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+2=0$
2. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+5=0$
3. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y+3=0$
4. $x^2-2y^2-2\sqrt{5}x-4\sqrt{2}y-2=0$

Q. The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y+1=0$, is

1. 
2. 
3. 
4. 

Q. The equation of a hyperbola, conjugate to the hyperbola $x^2+3xy+2y^2+2x+3y+1=0$, is

1. $x^2+3xy+2y^2+2x+3y+1=0$
2. $x^2+3xy+2y^2+2x+3y+2=0$
3. $x^2+3xy+2y^2+2x+3y+3=0$
4. $x^2+3xy+2y^2+2x+3y+4=0$