Question: The foci of a hyperbola are \((-5, -3)\) and \((9, -3)\). For any point on the hyperbola, the…

The foci of a hyperbola are \((-5, -3)\) and \((9, -3)\). For any point on the hyperbola, the absolute value of the difference of its distances from the foci is 10. Find the standard equation of the hyperbola.

Solution

The problem describes a hyperbola with foci at \((-5, -3)\) and \( (9, -3) \), and the absolute value of the difference of the distances from the foci is 10. We need to find the standard equation of the hyperbola. Determine the center of the hyperbola: The center \((h, k)\) is the midpoint of the segment joining the foci. \[ h = \frac{-5 + 9}{2} = 2 \] \[ k = \frac{-3 + (-3)}{2} = -3 \] So, the center is \((2, -3)\). Determine the distance between the foci: \[ 2c = \sqrt{(9 + 5)^2 + (-3 + 3)^2} = \sqrt{14^2} = 14 \] Therefore, \(c = 7\). Given that the absolute difference of distances is 10: \[ 2a = 10 \] Thus, \(a = 5\). Using the relationship \(c^2 = a^2 + b^2\): \[ c^2 = a^2 + b^2 \] \[ 7^2 = 5^2 + b^2 \] \[ 49 = 25 + b^2 \] \[ b^2 = 24 \] The standard equation of a hyperbola with a horizontal transverse axis is: \[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \] Substitute the values for \(h\), \(k\), \(a^2\), and \(b^2\): \[ \frac{(x - 2)^2}{25} - \frac{(y + 3)^2}{24} = 1 \] This is the standard equation of the hyperbola.