Circles
Centre and Radius from Tangent Lines and Normal
nta_pyq_2023_apr
Grade 11
Question:
Let the centre of a circle $C$ be $(\alpha,\beta)$ and its radius $r<8$. Let $3x+4y=24$ and $3x-4y=32$ be two tangents and $4x+3y=1$ be a normal to $C$. Then $(\alpha-\beta+r)$ is equal to
Step-by-Step Solution
Key Concept: Equate distances from centre $(\alpha,\beta)$ to both tangent lines: $\frac{|3\alpha+4\beta-24|}{5}=\frac{|3\alpha-4\beta-32|}{5}$. Also, $(\alpha,\beta)$ lies on normal $4x+3y=1$.
$\beta=-1,\ \alpha=1,\ r=5$. $\alpha-\beta+r=1-(-1)+5=7$.
Correct Answer: 1