Circles
Circle Touching Coordinate Axes — Distance Properties
nta_pyq_2023_apr
Grade 11
Question:
A circle passing through the point $P(\alpha,\beta)$ in the first quadrant touches the two coordinate axes at the points $A$ and $B$. The point $P$ is above the line $AB$. The point $Q$ on the line segment $AB$ is the foot of perpendicular from $P$ on $AB$. If $PQ$ is equal to $11$ units, then the value of $\alpha\beta$ is _______.
Step-by-Step Solution
Key Concept: A circle touching both axes has equation $(x-r)^2+(y-r)^2=r^2$. The chord $AB$ has equation $x+y=r$. Use the foot-of-perpendicular formula to find $Q$, then compute $PQ$.
Circle: $(x-r)^2+(y-r)^2=r^2$. $P(\alpha,\beta)$ on circle: $(\alpha-r)^2+(\beta-r)^2=r^2$. $PQ=\frac{|r-\alpha-\beta|}{\sqrt{2}}\cdot\sqrt{2}=|r-\alpha-\beta|/\sqrt{2}\cdot\sqrt{2}$... After calculation, $(r-\alpha-\beta)^2=242\Rightarrow 2\alpha\beta=242\Rightarrow\alpha\beta=121$.
Correct Answer: 121