For a hyperbola, the relationship between a, b, and c is c² = a² + b². We can substitute the value of c and the expression for b² from the previous steps into this equation to form a quadratic equation in terms of a.

For a hyperbola, the relationship between a, b, and c is c² = a² + b². We can substitute the value of c and the expression for b² from the previous steps into this equation to form a quadratic equation in terms of a.

Now that we have the value of a, we can calculate a². Then, using the relationship b² = 18a from step 2, we can find the value of b².

Since the foci are on the y -axis, the standard equation of the hyperbola is (y²)/(a²) - (x²)/(b²) = 1. We substitute the calculated values of a² and b² into this equation to get the final equation of the hyperbola.

Since the foci are on the y -axis, the standard equation of the hyperbola is (y²)/(a²) - (x²)/(b²) = 1. We substitute the calculated values of a² and b² into this equation to get the final equation of the hyperbola.

Find the equation of the hyperbola where foci are (0, +-6) and the length of the latus rectum is 36.

Answer: The equation of the hyperbola is $\frac{y^2}{198 - 54\sqrt{13}} - \frac{x^2}{-162 + 54\sqrt{13}} = 1$ .

Step-by-step solution

Step 1: Identify key parameters from foci

The given foci of the hyperbola are $(0, \pm 6)$ . For a hyperbola with foci on the $y$ -axis, the standard form of the foci is $(0, \pm c)$ . By comparing these, we can determine the value of $c$ .

Step 2: Use latus rectum length to find a relationship between a and b

The length of the latus rectum for a hyperbola is given by the formula $\frac{2b^2}{a}$ . We are given that the length of the latus rectum is $36$ . We can set up an equation using this information to find a relationship between $a^2$ and $b^2$ .

Step 3: Apply the relationship between a, b, and c

For a hyperbola, the relationship between $a$ , $b$ , and $c$ is $c^2 = a^2 + b^2$ . We can substitute the value of $c$ and the expression for $b^2$ from the previous steps into this equation to form a quadratic equation in terms of $a$ .

Step 4: Solve the quadratic equation for a

We solve the quadratic equation $a^2 + 18a - 36 = 0$ for $a$ using the quadratic formula. Since $a$ represents a length, it must be a positive value. We select the positive root from the two solutions.

Step 5: Calculate $a^2$ and $b^2$

Now that we have the value of $a$ , we can calculate $a^2$ . Then, using the relationship $b^2 = 18a$ from step 2, we can find the value of $b^2$ .

Step 6: Write the equation of the hyperbola

Since the foci are on the $y$ -axis, the standard equation of the hyperbola is $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ . We substitute the calculated values of $a^2$ and $b^2$ into this equation to get the final equation of the hyperbola.

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