Circles
Circles Touching Axes with Chord on Line
nta_pyq_2023_apr
Grade 11
Question:
Two circles in the first quadrant of radii $r_1$ and $r_2$ touch the coordinate axes. Each of them cuts off an intercept of $2$ units with the line $x+y=2$. Then $r_1^2+r_2^2-r_1 r_2$ is equal to ____.
Step-by-Step Solution
Key Concept: A circle touching both axes in Q1 has centre $(r,r)$ and equation $(x-r)^2+(y-r)^2=r^2$. The distance from centre to $x+y=2$ equals $\sqrt{r^2-1}$ when the chord length is $2$.
$r^2-4r+3=0\Rightarrow r_1=1,\ r_2=3$. $r_1^2+r_2^2-r_1r_2=7$.
Correct Answer: 7