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.

The length of the latus-rectum of a hyperbola is given by the formula (2b²)/(a). Substituting the values of b=12 and a=5, we can find the length of the latus-rectum.

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:

Answer: The length of the latus-rectum is $\frac{288}{5}$ .

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 ( $F_1$ and $F_2$ ) 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 $c^2 = a^2 + b^2$ . We have $c=13$ and $a=5$ , so we can solve for $b^2$ 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 $\frac{2b^2}{a}$ . Substituting the values of $b=12$ and $a=5$ , we can find the length of the latus-rectum.

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Let the foci of a hyperbola be (1,14)(1, 14)(1,14) and (1,−12)(1, -12)(1,−12). If it passes through the point (1,6)(1, 6)(1,6), then the length of its latus-rectum is: