Find the equation of the hyperbola where foci are (0,±12) and the length of the latus rectum is 36
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Step-by-Step Solution
Step 1: Identify Foci and Standard Form
The given foci are (0,±12). For a hyperbola, the foci are at (0,±c) for a vertical hyperbola, or (±c,0) for a horizontal hyperbola. Since the x -coordinate is 0, the foci are on the y -axis, indicating a vertical hyperbola. Therefore, c=12. The standard equation for a vertical hyperbola centered at the origin is a2y2−b2x2=1.
Step 2: Use Latus Rectum Length
The length of the latus rectum for a hyperbola is given by the formula a2b2. We are given that this length is 36. Setting up the equation, we get a2b2=36. Simplifying this equation, we find b2=18a.
Step 3: Relate a, b, and c
For a hyperbola, the relationship between a, b, and c is c2=a2+b2. We know c=12 and we found b2=18a. Substituting these values into the equation, we get 122=a2+18a, which simplifies to 144=a2+18a. Rearranging this into a standard quadratic form gives a2+18a−144=0.
Step 4: Solve for 'a'
We solve the quadratic equation a2+18a−144=0 for a. Factoring the quadratic, we find (a+24)(a−6)=0. This gives two possible values for a: a=−24 or a=6. Since a represents a distance, it must be positive, so we choose a=6.
Step 5: Calculate 'b²'
Now that we have a=6, we can find b2 using the relationship b2=18a that we derived from the latus rectum length. Substituting a=6 into this equation, we get b2=18(6), which simplifies to b2=108.
Step 6: Write the Equation of the Hyperbola
We have determined that a=6 (so a2=36) and b2=108. Substitute these values into the standard equation for a vertical hyperbola: a2y2−b2x2=1. This gives us the final equation of the hyperbola: 36y2−108x2=1.