AB is the chord of contact of tangents drawn from a point $(6,8)$ to the circle $\displaystyle x^{2}+y^{2}= r^{2}.$ If the area of the triangle $PAB$ be maximum, then radius $r$ of the circle is ?

For a point (x1, y1) outside a circle x^2 + y^2 = r^2, the chord of contact is xx1 + yy1 = r^2. Here, 6x + 8y = r^2. The distance from the origin to this line is d = r^2 / sqrt(6^2 + 8^2) = r^2 / 10. The length of the chord AB is 2 * sqrt(r^2 - d^2). Area = 1/2 * base * height = 1/2 * (2 * sqrt(r^2 - d^2)) * (sqrt(x1^2 + y1^2) - d) = sqrt(r^2 - d^2) * (10 - d). Substituting d = r^2/10, Area = sqrt(r^2 - r^4/100) * (10 - r^2/10) = (r/10) * sqrt(100 - r^2) * (10 - r^2/10) = (r/100) * (100 - r^2)^(3/2). Maximize by setting derivative to 0, which occurs at r = 5.