Solve C:x^2/a^2-y^2/b^2=1

$$C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$

Solve for C

$$C=\left(\frac{x}{ab}\right)^{2}\left(y^{2}+b^{2}\right)$$

$$x\neq 0\text{ and }a\neq 0\text{ and }b\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by b^{2}.

$$b^{2}a^{2}x^{-2}C-y^{2}=b^{2}$$

Add y^{2} to both sides.

$$b^{2}a^{2}x^{-2}C=b^{2}+y^{2}$$

The equation is in standard form.

$$\frac{a^{2}b^{2}}{x^{2}}C=y^{2}+b^{2}$$

Divide both sides by b^{2}a^{2}x^{-2}.

$$\frac{\frac{a^{2}b^{2}}{x^{2}}Cx^{2}}{a^{2}b^{2}}=\frac{\left(y^{2}+b^{2}\right)x^{2}}{a^{2}b^{2}}$$

Dividing by b^{2}a^{2}x^{-2} undoes the multiplication by b^{2}a^{2}x^{-2}.

$$C=\frac{\left(y^{2}+b^{2}\right)x^{2}}{a^{2}b^{2}}$$

Divide b^{2}+y^{2} by b^{2}a^{2}x^{-2}.

$$C=\frac{x^{2}\left(y^{2}+b^{2}\right)}{\left(ab\right)^{2}}$$

Solve for a

$$a=\frac{\sqrt{\frac{y^{2}+b^{2}}{C}}|x|}{b}$$

$$a=-\frac{\sqrt{\frac{y^{2}+b^{2}}{C}}|x|}{b}\text{, }b\neq 0\text{ and }C>0\text{ and }x\neq 0$$