The sum of all the values of λ for which the set x , y : x 2 + y 2 - 6 x + 4 y = 12 ∩ x , y : 4 x + 3 y = λ contains exactly one element is

Step-by-step solution

For exactly one element, the line should be a tangent to the circle Given line L : 4 x + 3 y = λ Circle S : x 2 + y 2 - 6 x + 4 y = 12 has centre 3 , - 2 , radius = 5 Now, L will be a tangent to the circle if the distance of it from the centre of the circle is equal to the radius ∴ p = r 4 3 + 3 - 2 - λ 5 = 5 ⇒ λ = 31 , - 19 ⇒ Sum of all the values of λ is equal to 12