Let the lines y + 2 x = 11 + 7 7 and 2 y + x = 2 11 + 6 7 be normal to a circle C : x - h 2 + y - k 2 = r 2 . If the line 11 y - 3 x = 5 77 3 + 11 is tangent to the circle C , then the value of 5 h - 8 k 2 + 5 r 2 is equal to ______.

Step-by-step solution

Equations of normal are y + 2 x = 11 + 7 7       . . . i 2 y + x = 2 11 + 6 7       . . . i i Now the center of the circle is point of intersection of the normals i.e. solving i   &   i i , we get the point of intersection as 8 7 3 , 11 + 5 7 3 ≡ h , k The equation of tangent is 11 y - 3 x = 5 77 3 + 11 The radius will be perpendicular distance of tangent from center i.e. r = 11 8 7 3 - 3 11 + 5 7 3 - 5 77 3 - 11 11 + 9 = 4 7 5 Hence 5 h - 8 k 2 + 5 r 2 = 816