Reorder <math><mstyle displaystyle="true"><mn>9</mn><mi>i</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>12</mn><mi>x</mi></mstyle></math> .

Find the standard form of the hyperbola.

Subtract <math><mstyle displaystyle="true"><mn>12</mn><mi>x</mi></mstyle></math> from both sides of the equation.

Divide each term by <math><mstyle displaystyle="true"><mn>9</mn><mi>i</mi></mstyle></math> to make the right side equal to one.

Simplify each term in the equation in order to set the right side equal to <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> . The standard form of an ellipse or hyperbola requires the right side of the equation be <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

This is the form of a hyperbola. Use this form to determine the values used to find the asymptotes of the hyperbola.

Match the values in this hyperbola to those of the standard form. The variable <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> represents the x-offset from the origin, <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> represents the y-offset from origin, <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> .

The asymptotes follow the form <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>±</mo><mfrac><mrow><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>h</mi><mo>)</mo></mrow></mrow><mrow><mi>a</mi></mrow></mfrac><mo>+</mo><mi>k</mi></mstyle></math> because this hyperbola opens left and right.

Simplify <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Simplify <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac><mi>x</mi><mo>+</mo><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> .

This hyperbola has two asymptotes.

The asymptotes are <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>x</mi></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>x</mi><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> .

Asymptotes: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>x</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>,</mo><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>x</mi><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>

Asymptotes: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mfrac><mrow><msqrt><mn>3</mn></msqrt><mi>x</mi></mrow><mrow><mn>6</mn></mrow></mfrac><mo>,</mo><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mi>x</mi><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math>

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