Find the equation of the hyperbola where foci are (0, +-6) and the length of the latus rectum is 36.
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Step-by-Step Solution
Step 1: Identify key parameters from foci
The given foci of the hyperbola are (0,±6). For a hyperbola with foci on the y -axis, the standard form of the foci is (0,±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 a2b2. 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 a2 and b2.
Step 3: Apply the relationship between a, b, and c
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.
Step 4: Solve the quadratic equation for a
We solve the quadratic equation a2+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 a2 and b2
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.
Step 6: Write the equation of the hyperbola
Since the foci are on the y -axis, the standard equation of the hyperbola is a2y2−b2x2=1. We substitute the calculated values of a2 and b2 into this equation to get the final equation of the hyperbola.