JEE Advanced 2017 Paper 1Question 1
For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?
Circle
Answer$2$
Solution
Centre $(-1,-2)$, $r^2=5+p$. Three common points: either it touches one axis and cuts the other, or it passes through the origin. Touching $x$-axis: $r=2$, $p=-1$ (cuts $y$-axis at two points: 3 total). Touching $y$-axis: $r=1$, $p=-4$ misses the $x$-axis (1 point only). Through origin: $p=0$ gives $(-2,0)$ and $(0,-4)$ besides the origin: 3 points. So 2 values.