Circles
Inscribed Circle — Three Tangent Lines
nta_pyq_2023_jan
Grade 11
Question:
Let $y=x+2$, $4y=3x+6$ and $3y=4x+1$ be three tangent lines to the circle $(x-h)^2+(y-k)^2=r^2$. Then $h+k$ is equal to:
Step-by-Step Solution
Key Concept: Lines: $y=x+2$ and $4y=3x+6\Rightarrow y=3x/4+3/2$ are parallel (slopes 1 and 3/4). Wait: slope of $y=x+2$ is 1, slope of $4y=3x+6$ is $3/4$, slope of $3y=4x+1$ is $4/3$. Check: $3/4\cdot4/3=1$, so last two are perpendicular.
$h+k=5$.
Correct Answer: 1