Definition of a Hyperbola

Q.

(Distributive laws) For any three sets A, B, C prove that:

I. $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
[Distirbutive law of union over intersection]

II. $A\cap \left[\right(B\cup C\left)\right]=(A\cap B)\cup (A\cap C)$
[Distributive law of intersection over union]

Q.

In a hyperbola e = $\frac{9}{4}$ and the distance between the directrices is 3. Then the length of Transverse axis is

1. 17/4

2. 27/2

3. 27/4

4. 27/8

Q.

The complex number $z_2$(2 + 3i) is rotated through an angle of ${90}^{\circ }$ anti-clockwise about $z_1$(1 + i). Find the new complex number $z_3$ obtained.

1. 1 + 2i

2. 1-2i

3. -1 + 2i

4. -1 -2i

Q. Find the coordinates of the foci, the vertices the eccentricity and the length of the latus rectum of the hyperbola $5y^2-9x^2=36$

Q. Latus rectum of a hyperbola is 8 and its conjugate axis is equal to half the distance between the foci. The eccentricity of the hyperbola is

1. $\frac{4}{3}$
2. 43
3. 23
4. 23

Q. The normal at an end of a latus rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through an end of the minor axis if

1. 
2. 
3. 
4. 

Q. Find the coordinates of the foci and the vertices the eccentricity and the length of the latus rectum of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$

Q. Find the direction in which a straight line must be drawn through the point $(1,2)$, so that its point of intersection with the line $x+y$ $=$ $4$ may be at a distance of $3$ units from this point.

Q.

Select the right options about a hyperbola.

1. Conjugate axis is the axis of the hyperbola that passes through the foci

2. A transverse axis always pass through the origin

3. The straight line through the centre which is perpendicular to the transverse axis doesn't meet hyperbola in real point.

4. Eccentricity is the ratio of 2 distances

Q. A straight line passing through the centre which is perpendicular to the transverse axis doesn't meet hyperbola in a real point.

1. False
2. True

Q. Find the coordinates of the foci, the vertices the eccentricity and the length of latus rectum of the hyperbola $16x^2-9y^2=576$