Question

A hyperbola has its center at (3, 4), a vertex at the point (9, 4), and the length of its latus rectum is 3 units. An extremity of the conjugate point axis is at the
(- 3, 4)
(4,4)
(3, 7)
(3,6) ​

Answer 1

SOLUTION

GIVEN

• A hyperbola has its center at (3, 4)

• Vertex at the point (9, 4)

• The length of its latus rectum is 3 units

TO CHOOSE THE CORRECT OPTION

An extremity of the conjugate point axis is at the

• (- 3, 4)

• (4,4)

• (3, 7)

• (3,6)

EVALUATION

Let the centre of the hyperbola

$\sf{ = ( \alpha , \beta )}$

Then the coordinates of the vertex are

$= \sf{( \alpha + a, \beta ) \: , ( \alpha - a , \beta )\: }$

Length of the latus rectum

$= \displaystyle \sf{ \frac{ 2{b}^{2} }{a} }$

So by the given condition

$\sf{ \alpha = 3 \: , \: \beta = 4}$

Also

$\sf{ \alpha + a = 9}$

$\implies \sf{3 + a = 9}$

$\implies \sf{a = 6}$

Also by the given condition

$\displaystyle \sf{ \frac{ 2{b}^{2} }{a} } = 3$

$\implies \: \displaystyle \sf{ \frac{ 2{b}^{2} }{6} = 3}$

$\implies \: \displaystyle \sf{ {b}^{2} = 9 }$

$\implies \: \displaystyle \sf{b = \pm \: 3}$

∴ The coordinates of the extremities of the conjugate point axis

$= \sf{ ( \alpha \: , \beta + b) \: \: and \: \: ( \alpha \: , \beta + b )}$

$= \sf{ ( 3 \: , 4 + 3) \: \: and \: \: ( 3 \: , 4 - 3 )}$

$= \sf{ ( 3 \: , 7) \: \: and \: \: ( 3 \: , 1 )}$

Hence an extremity of the conjugate point axis is at the point ( 3, 7 )

Graph :

In the graph

• A and A' are vertex

• C is the centre

• B and B' are extremities of the conjugate point axis

FINAL ANSWER

Hence the correct option is ( 3, 7 )

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Find the center and radius of circle represented by

xsquare +ysquare+6x-4y+4=0

https://brainly.in/question/28987139

2. The center and radius of the circle given by x2+y2-4x-5=0

https://brainly.in/question/29016682