Question

Writing an equation of a hyperbola give me the Foci and the isotopes

Answer 1

Step 1

The general equation of the hyperbola is given below

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

Where (h,k) is the center and a and b are the length of semi-major and semi-minor

Step 2

K = 0 and h = 6

`\begin{gathered} (h\text{ + 6\rparen}^2\text{ = a}^2\text{ + b}^2 \\ (h\text{ - 6\rparen}^2\text{ = a}^2\text{ + b}^2 \\ We\text{ know that }(b)/(a)\text{ = 1} \\ b\text{ = a} \end{gathered}`

Step 3

`\begin{gathered} a^2\text{ + b}^2\text{ = 6}^2 \\ 2a^2\text{ = 36} \\ a^2\text{ = }(36)/(2) \\ a^2\text{ = 18} \\ a\text{ = }√(18) \\ a\text{ = 3}√(2) \\ \text{b = 3}√(2) \end{gathered}`

Step 4

`\begin{gathered} \text{The standard equation of a parabola is }(x^2)/((3√(2))^2)\text{ - }(y^2)/((3√(2))^2)\text{ = 1} \\ or \\ (x^2)/(18)\text{ - }(y^2)/(18)\text{ = 1} \end{gathered}`