Circles
Circle Through Points Touching Another Circle
nta_pyq_2024_apr
Grade 11
Question:
Let the centre of a circle, passing through the points $(0,0)$, $(1,0)$ and touching the circle $x^2+y^2=9$, be $(h,k)$. Then for all possible values of the coordinates of the centre $(h,k)$, $4(h^2+k^2)$ is equal to ________.
Step-by-Step Solution
Key Concept: Circle through $(0,0)$ and $(1,0)$: centre on perpendicular bisector of segment $(0,0)$–$(1,0)$, i.e. $h=1/2$. Radius $r=\sqrt{h^2+k^2}$. Touches $x^2+y^2=9$ (internally or externally): $\sqrt{h^2+k^2}=3\pm r$... since it passes through origin, $r=\sqrt{h^2+k^2}$. Internal tangency: $3-r=r\Rightarrow r=3/2$.
$h=1/2$, $k^2=2$. $4(h^2+k^2)=4(1/4+2)=9$.
Correct Answer: 9