Let C(3, 4) be the center of a circle. Tangents drawn from the origin O touch the circle at distinct points P(a₁, b₁) and Q(a₂, b₂) . If the area of the quadrilateral OPCQ is 12 and the radius of the circle is strictly less than the length of the tangent from O to the circle, then the value of a₁^2 + a₂^2 + b₁^2 + b₂^2 is equal to

Step-by-step solution

Let the origin be O(0, 0) . The square of the distance of the center C(3, 4) from the origin is given by OC^2 = 3^2 + 4^2 = 25 . The tangents from O touch the circle at P(a₁, b₁) and Q(a₂, b₂) . The lengths of the tangents from an external point are equal, so OP = OQ . Also, the radius is perpendicular to the tangent at the point of contact, so OPC = OQC = 90^ . The area of the quadrilateral OPCQ is the sum of the areas of the two congruent right-angled triangles OPC and OQC . Area of OPCQ = 2 ( 1 2 OP CP ) = OP r