Circles
Circle
Allen Star Batch
Grade 11
Question:
Consider two circles $S_1$ and $S_2$ (externally touching) having centres at points $A$ and $B$ whose radii are 1 and 2 respectively. A tangent to circle $S_1$ from point $B$ touches the circle $S_1$ at point $C$. $D$ is chosen on circle $S_2$ so that $AC$ is parallel to $BD$ and two segments $BC$ and $AD$ do not intersect. Segment $AD$ intersect the circle $S_1$ at $E$. The line through $B$ and $E$ intersects the circle $S_1$ at another point $F$.
The length of segment $EF$ is $\frac{2\sqrt{3}}{3}$
The area of triangle $ABD$ is $2\sqrt{2}$
The length of the segment $DE$ is $2$
$ABD$ is a triangle of perimeter $2\sqrt{3}$
Step-by-Step Solution
Key Concept: Use slope relationships and coordinate geometry to find intersection points of lines with circles.
In triangle $ABC$ with $\tan\theta = 2\sqrt{2}$ and $AG = 2\sqrt{2}$, the slope of $BD$ is $m_{BD} = 2\sqrt{2}$. Using point $A$ and the given conditions, we find $D = \left(\frac{7}{3}, -\frac{4\sqrt{2}}{3}\right)$. The slope of line $AE$ equals the slope of $AD$, which is $\frac{-4\sqrt{2}}{7}$. Solving the intersection of line $BE$ with circle $S_1$ using $y = \frac{\sqrt{2}}{5}(x-3)$ gives $F = \left(\frac{1}{3}, -\frac{2\sqrt{2}}{3}\right)$.
Correct Answer: 1,2,3