The diagram shows the circle with equation $$\(x^{2}+y^{2}-6 x+4 y-27=0\)$$ and the tangent to the circle at the point $$\(P(5,4)\)$$. Points $$\(Q\)$$ and $$\(R\)$$ also lie on the circle, such that $$\(P Q R\)$$ is an equilateral triangle. Find the exact area of triangle $$\(P Q R\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w21_qp_12 Year：2021 Question No：12(b)
Answer:
Radius of circle \(=\sqrt{40}\),
Area of \(\Delta C R Q=\frac{1}{2} \times(\text { their } r)^{2} \sin 120\left[=\frac{1}{2} \times 40 \times \frac{\sqrt{3}}{2}\right]\)
OR
Area of \(\triangle C Q X=\frac{1}{2} \times \sqrt{40} \cos 30 \times \sqrt{40} \cos 60 \mathrm{OE}\left[=\frac{1}{2} \times \sqrt{30} \times \sqrt{10}\right]\)
\(\mathrm{OR}\)
Area of circle \(-3 \times\) Area of segment \(=40 \pi-3 \times\left(40 \frac{\pi}{3}-10 \sqrt{3}\right)\)
OR
\(Q R=\sqrt{120}\) or \(2 \sqrt{30}\) and area \(=\frac{1}{2} Q R^{2} \sin 60\)
\(30 \sqrt{3}\)
Area of \(\Delta C R Q=\frac{1}{2} \times(\text { their } r)^{2} \sin 120\left[=\frac{1}{2} \times 40 \times \frac{\sqrt{3}}{2}\right]\)
OR
Area of \(\triangle C Q X=\frac{1}{2} \times \sqrt{40} \cos 30 \times \sqrt{40} \cos 60 \mathrm{OE}\left[=\frac{1}{2} \times \sqrt{30} \times \sqrt{10}\right]\)
\(\mathrm{OR}\)
Area of circle \(-3 \times\) Area of segment \(=40 \pi-3 \times\left(40 \frac{\pi}{3}-10 \sqrt{3}\right)\)
OR
\(Q R=\sqrt{120}\) or \(2 \sqrt{30}\) and area \(=\frac{1}{2} Q R^{2} \sin 60\)
\(30 \sqrt{3}\)
Knowledge points:
1.1.1 carry out the process of completing the square for a quadratic polynomial and use a completed square form
1.3.3 understand that the equation represents the circle with centre and radius (Including use of the expanded form.)
1.4.2 use the formulae in solving problems concerning the arc length and sector area of a circle (Including calculation of lengths and angles in triangles and areas of triangles.)
Solution:
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