Let P a 1 , b 1 and Q a 2 , b 2 be two distinct points on a circle with center C ( 2 , 3 ) . Let O be the origin and O C be perpendicular to both C P and C Q . If the area of the triangle O C P is 35 2 , then a 1 2 + a 2 2 + b 1 2 + b 2 2 is equal to __________

Step-by-step solution

Given, P a 1 ,   b 1 and Q a 2 ,   b 2 be two distinct points on a circle with center C ( 2 , 3 ) , And O be the origin and O C be perpendicular to both C P and C Q , So, P C Q will be a straight line Now on plotting the diagram we get, So, O C = 2 2 + 3 2 = 5 Now let C P = C Q = I Now by using area of triangle O C P = 1 2 × O C × I we get, 35 2 = 1 2 × 5 × I ⇒ I = 7 And O P = O Q = O C 2 + I 2 = 12 So, a 1 2 + b 1 2 + a 2 2 + b 2 2 = O P 2 + O Q 2 = 12 + 12 = 24