Circles
Circle
Allen Star Batch
Grade 11
Question:
The circle '$S$' touches the sides $AB$ and $AD$ of the rectangle $ABCD$ and cuts the side $DC$ at single point $F$ and the side $BC$ at a single point $E$. If $|AB| = 32, |AD| = 40$ and $|BE| = 1$
The angle between pair of tangents drawn form the point $D$ to the circle '$S$' is $\pi - \tan^{-1}\left(\frac{15}{8}\right)$
The Area of trapezium $AFCB$ is $1180$ sq.units
The radius of circle is $25$ units
The angle between pair of tangents drawn form the point $D$ to the circle '$S$' is $\pi - 2\tan^{-1}\left(\frac{15}{8}\right)$
Step-by-Step Solution
Key Concept: Substitute a known point into the circle equation to determine the radius, then use coordinate geometry to find areas.
The equation of the circle is $x^2 + y^2 - 2rx - 2ry + r^2 = 0$. Substituting point $(32, 1)$ gives $r^2 - 66r + 1025 = 0$, yielding $r = 25$ or $r = 41$. Since $r = 41$ is rejected, we have $r = 25$. Using $\tan^{-1}\left(\frac{25}{15}\right) = \tan^{-1}\left(\frac{5}{3}\right)$ and angle subtraction, the area of triangle $AFCB = \frac{1}{2}(40)(32 + 27) = 1180$.
Correct Answer: 1,2,3