Class 11

Math

3D Geometry

Conic Sections

Find the equation of the hyperbola satisfying the give conditions: Foci $(0,± 10 )$

passing through $(2,3)$

Therefore, the equation of the hyperbola is of the form $a_{2}y_{2} −b_{2}x_{2} =1$

Since the foci are $(0,± 10 )⇒ae=c=10 $

We know that $a_{2}+b_{2}=c_{2}$

$∴$ $a_{2}+b_{2}=10$

$⇒b_{2}=10−a_{2}...(1)$

Since the hyperbola passes through point $(2,3)$

$a_{2}9 −b_{2}4 =1 ...(2)$

From equation (1) and (2), we obtain

$a_{2}9 −(10−a_{2})4 =1$

$⇒9(10−a_{2})−4a_{2}=a_{2}(10−a_{2})$

$⇒90−9a_{2}−4a_{2}=10a_{2}−a_{4}$

$⇒a_{4}−23a_{2}+90=0$

$⇒$$a_{4}−18a_{2}−5a_{2}+90=0$

$⇒a_{2}(a_{2}−18)−5(a_{2}−18)=0$

$⇒$$(a_{2}−18)(a_{2}−5)=0$

$⇒$$a_{2}=18or5$

In hyperbola $c>a$, i.e., $c_{2}>a_{2}$

$∴a_{2}=5$

$⇒$$b_{2}=10−a_{2}=10−5=5$

Thus the equation of the hyperbola is $5y_{2} −5x_{2} =1$