Let (x, y) be satisfy the curve x^2+y^2-6 x-8 y+21=0 and m₁ and m₂ are minimum and maximum values of ( x y ) . If m₁^2+m₂^2= p q where p, q N , then find the least value of (p+q) .

Step-by-step solution

x y = 1 m Let y=m x be the tangent Applying, aligned & p=r & | 3 m-4 1+m^2 |=2 & 9 m^2-24 m+16=4+4 m^2 & 5 m^2-24 m+12=0 & 12 m^2-24 m+5=0 & m₁ m₁ aligned aligned (m₁+m₂ )^2-2 m₁ m₂ & =4-2 ( 5 12 ) 19 6 p q p+q & =25 aligned