An ellipse in the first quadrant is tangent to both the \(x\)-axis and the \(y\)-axis. The coordinates of its foci are \((8,12)\), and \((v,12)\). Compute \(v\).
Solution
Answer: \(18\)
Use the ellipse definition: the sum of distances to the two foci is the same at both tangent points. The tangent point on the \(x\)-axis lies on the perpendicular bisector of the foci, giving
\[8+v=2\sqrt{\left(\frac{|8-v|}{2}\right)^2+12^2}.\]
Squaring both sides yields \(64+16v+v^2=v^2-16v+640\). Hence \(32v=576\), so \(v=18\).