Question

Find hyperbola equation. center (0,0) vertex (-2,0) focus (-5,0)

Answer 1

`\frac{ {x}^(2) }{4} - \frac{ {y}^(2) }{21} = 1`

`\frac{(x - h)^(2) }{ {a}^(2) } - \frac{(y - k) ^(2) }{ {b}^(2) } = 1 \\`

a= (–2, 0) ; Center =(0,0)

`distance = \sqrt{(x2 - x1)^(2) + (y2 - y1) ^(2) } \\ a = \sqrt{(( - 2) - 0)^(2) + (0 - 0) ^(2) } \\ a = \sqrt{ {2}^(2) } \\ a = 2`

C = (–5,0) ; Center =(0,0)

`distance = \sqrt{(x2 - x1) ^(2) + (y2 - y1) ^(2) } \\ c = \sqrt{(( - 5) - 0)^(2) + (0 - 0) ^(2) } \\ c = \sqrt{ {5}^(2) } \\ c = 5`

C²= a²+ b²

(5)²= (2)² + b²

b²= 25–4 —> b² = 21

`b = + √(21) , - √(21)`

`m = (y2 - y1)/(x2 - x1) = (0 - 0)/(0 - ( -5 )) = 0`

`\frac{(x - h)^(2) }{ {a}^(2) } - \frac{(y - k) ^(2) }{ {b}^(2) } = 1 \\`

`\frac{(x - 0)^(2) }{ {2}^(2) } - \frac{(y - 0) ^(2) }{ { √(2) }^(2) } = 1 \\`

`\frac{ {x}^(2) }{4} - \frac{ {y}^(2) }{21} = 1`

I hope I helped you^_^