Let the foci of a hyperbola be (1,14) and (1,−12). If it passes through the point (1,6), then the length of its latus-rectum is:
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Step-by-Step Solution
Step 1: Find the center and distance between foci
The center of the hyperbola is the midpoint of the segment connecting the two foci. The distance between the foci is 2c, where c is the distance from the center to each focus. Given the foci are (1,14) and (1,−12), we can calculate these values.
Step 2: Calculate 'c' and 'a'
The center is (1,1). The distance between the foci is 26, so c=13. For any point P on the hyperbola, the absolute difference of its distances from the two foci (F1 and F2) is equal to 2a, where a is the distance from the center to each vertex.
Step 3: Calculate 'a' using the given point
The hyperbola passes through the point (1,6). We use the distance formula to find the distances from this point to each focus. The absolute difference of these distances gives us 2a. This calculation yields a=5.
Step 4: Calculate 'b'
For a hyperbola, the relationship between a, b, and c is given by c2=a2+b2. We have c=13 and a=5, so we can solve for b2 and then b.
Step 5: Calculate the length of the latus-rectum
The length of the latus-rectum of a hyperbola is given by the formula a2b2. Substituting the values of b=12 and a=5, we can find the length of the latus-rectum.