Question

the line x-y+3=0 touches the hyperbola whose foci are (⁺₋√41,0).find the equation of the hyperbola​

Answer 1

Answer:

Since, the foci of the given hyperbola are F

′

(0,−

10

) and F(0,

10

) which lie on the X-axis and mid-point of the segment F

′

F is (0,0). Therefore origin is the centre of the hyperbola and its transverse axis lies along Y-axis, hence the equation of the hyperbola can be taken as

a

2

y

2

−

b

2

x

2

=1 .........(i)

As the foci are (0,±

10

)

So, ae=

10

We know that

b

2

=a

2

e

2

−a

2

⇒b

2

=10−a

2

Answer 2

x 2 −2y 2=18

x+y=a

y=(x−a)

⇒x 2 −2(x−a) 2 =18

⇒−x +4ax−2a 2=18

⇒x 2−4ax+2(a 2−18)=0

Since line touches the hyperbola. So only one value of x. Hence D=0

⇒16a 2 −8(a 2 −18)=0

a 2 =9

∣a∣=3.