Question

find the equation for a hyperbola centered at (0,0),with foci at (-square root 34,0) and (square root 34,0) and vertices at (3,0) and (-3,0)

Answer 1

`(x^2)/(9)-(b^2)/(25)=1`

Explanation:

Since, we know that,

The equation of hyperbola,

`((x-h)^2)/(a^2)-((y-k)^2)/(b^2)=1`

Where, (h,k) is the center of the hyperbola,

Here, h = k = 0,

So, the equation of hyperbola is,

`(x^2)/(a^2)-(y^2)/(b^2)=1-----(1)`

Also, a = distance between center and vertices of hyperbola,

Given, vertices = (±3,0)

`a=√((0-3)^2+(0-0)^2)=√(9)=3`

`\implies a^2 = 9`

Now, foci, c= (±√34,0),

Since, foci of hyperbola (1) is,

`(\pm√(a^2+b^2),0)`

`\implies a^2+b^2= (√(34))^2`

`\implies 3^2 + b^2 = 34`

`\implies b^2 = 34 - 9 = 25`

Hence, the equation of the given hyperbola is,

`(x^2)/(9)-(b^2)/(25)=1`

Answer 2

THe hyperbola is horizontally oriented.
c^2 = a^2 + b^2
34 = 9 + b^2
b^2 = 25
b = 5
x^2/9 - y^2/25 = 1

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