UPSC NDA Math Model Paper 2

Let the circle f(x,y)=x2+y2+10x−12y+51=0 and the line g(x,y)=3y+x−3=0 intersect at a point P(a, b).
∴ g(a, b) = 0
3b + a - 3 = 0
⇒ a = 3 - 3b ... (1)
And, f(a, b) = 0
⇒a2+b2+10a−12b+51=0
Using equation (1), we get:
⇒(3−3b)2+b2+10(3−3b) −12b+51=0
⇒9−18b+9b2+b2+30−30b −12b+51=0
⇒10b2−60b+90=0
⇒b2−6b+9=0
⇒(b−3)2=0
⇒b=3
And, using equation (1):
a=3−3×3=3−9=−6.
Therefore, the given curves intersect at point (-6, 3).