Circles
Circle Tangent to Line — Finding Parameters
nta_pyq_2024_jan
Grade 11
Question:
Consider a circle $(x-\alpha)^2+(y-\beta)^2=50$, where $\alpha,\beta>0$. If the circle touches the line $y+x=0$ at the point $P$, whose distance from the origin is $4\sqrt{2}$, then $(\alpha+\beta)^2$ is equal to
Step-by-Step Solution
Key Concept: Circle touches $x+y=0$, so distance from centre $=(\alpha+\beta)/\sqrt{2}=\sqrt{50}=5\sqrt{2}\Rightarrow\alpha+\beta=10\Rightarrow(\alpha+\beta)^2=100$.
Distance from $(\alpha,\beta)$ to $x+y=0$: $\frac{\alpha+\beta}{\sqrt{2}}=5\sqrt{2}\Rightarrow\alpha+\beta=10$. $(\alpha+\beta)^2=100$.
Correct Answer: 100