KC Sinha Mathematics Solution Class 11 Chapter 25 Conic Section: Hyperbola Exercise 25.1

 Exercise 25.1

Page no  25.25

Type 1

Question 1

Find the equation to the hyperbola for which eccentricity is 2. One Focus is (2,2) and the corresponding directrix is x+ y = 9

 Question 2

The equation of the directrix of a hyperbola is $x-y+3=0$, its focus is $(-1,1)$ and eccentricity is 3. Find the equation of the hyperbola.

 Question 3

Find the equation of the hyperbola whose conjugate axis is 5 and the distance between the foci is 13, taking transverse and conjugate axes along $x$ and $y^{\text {-axes. }}$.

 Question 4

 Find the equation of the hyperbola having foci $(0, \pm 4)$ and transverse axis of length $6 .$

 Question 5

Find the equation of the hyperbola with vertices at $(0, \pm 6)$ and eccentricity $5 / 3$.

 Question 6

Find the equation of the hyperbola having vertices $(0, \pm 5)$ and foci $(0, \pm 8)$.

Page no  25.26

 Question 7

Find the equations of the hyperbola having

(i) vertices $(0, \pm 3)$ and foci $(0, \pm 5)$

(ii) vertices $\left(0, \pm \frac{\sqrt{11}}{2}\right)$ and foci $(0, \pm 3)$

(iii) vertices $(\pm 2,0)$ and foci $(\pm 3,0)$

 Question 8

Find the equation of the hyperbola having

(i) foci $(\pm 4,0)$ and the length of latus rectum 12

(ii) foci $(0, \pm 12)$ and the length of latus rectum 36

(iii) foci $(\pm 3 \sqrt{5}, 0)$ and the length of latus rectum 8

 Question 9

Find the equation of the hyperbola having

(i) foci $(0, \pm 13)$ and length of conjugate axis 24

(ii) foci $(\pm 5,0)$ and length of transverse axis 8

Question 10

Find the equation of the hyperbola having vertices $(\pm 7,0)$ and $e=\frac{4}{3}$.

Question 11

Find the equation of the hyperbola whose vertices are $(\pm 6,0)$ and one of the directrices is $x=4$.

Question 12

Find the equation to the hyperbola if

(i) the distance between the foci is 9 and eccentricity is $\sqrt{3}$, taking transverse and conjugate axes along $x$ and $y$-axes respectively.

(ii) the foci are at $(6,4)$ and $(-4,4)$ and eccentricity is 2 .

Question 13

The co-ordinates of the foci of a hyperbola are $(\pm 6,0)$ and its latus rectum is of 10 units. Find the equation of the hyperbola.

Question 14

Find the equation to the hyperbola referred to its axes as co-ordinate axes whose conjugate axis is 7 and passes through the point $(3,-2)$.

Type 2

Question 15

In the hyperbola $4 x^{2}-9 y^{2}=36$, find the axes, the co-ordinates of the foci, the eccentricity, and the latus rectum.

Question 16

Find the coordinates of the vertices, the foci, the eccentricity and the equations of directrices of the hyperbola $4 x^{2}-25 y^{2}=100$

Question 17

Find the coordinates of the vertices, the foci, the eccentricity and the equations of the directrices of the following hyperbolas :

(i) $3 x^{2}-2 y^{2}=1$

(ii) $16 x^{2}-9 y^{2}=144$

(iii) $16 y^{2}-4 x^{2}=1$

(iv) $y^{2}-16 x^{2}=16$

Question 18

Find the foci, vertices, eccentricity and length of latus rectum of the following hyperbolas :

(i) $16 x^{2}-9 y^{2}=576$

(ii) $49 y^{2}-16 x^{2}=784$

(iii) $3 y^{2}-x^{2}=27$

(iv) $5 y^{2}-9 x^{2}=36$

(v) $\frac{y^{2}}{4}-\frac{x^{2}}{9}=1$

Question 19

Find the center, eccentricity, foci and directrices of the hyperbola

$16 x^{2}-9 y^{2}+32 x+36 y-164=0$

Page no  25.27

Question 20

Show that the equation

$9 x^{2}-16 y^{2}-18 x-64 y-199=0$

Represent a hyperbola for this hyperbola, find the length of axes, Eccentricity center, foci , vertices, latus rectum and directrices

Question 21

Fad the length of axes, the eccentricity, center, foci and latus rectum of the hyperbola $16 x^{2}-3 y^{2}-32 x-12 y-44=0$

Type 3

Question 22

The hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ passes through the point of intersection of the lines $7 x+13 y-87=0$ and $5 x-8 y+7=0$ and its latus rectum is $\frac{32 \sqrt{2}}{5}$. Find a and b.

Question 23

 $P V$ is the ordinate of any point $P$ on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ if $Q$ divides $A P$ in the ratio $a^{2}: b^{2}$, show that $N Q$ is perpendicular to $A^{\prime} P$ where $A A^{\prime}$ is the transverse axis of the hyperbola.

Question 24

1. Prove that the locus of the point of intersection of the lines $\sqrt{3} x-y-4 \sqrt{3} k=0$ and $\sqrt{3} k x+k y-4 \sqrt{3}=0$ for different values of $k$ is a hyperbola whose eccentricity is 2 .