The diagram shows a sector $$\(A B C\)$$ which is part of a circle of radius $$\(a\)$$. The points $$\(D\)$$ and $$\(E\)$$ lie on $$\(A B\)$$ and $$\(A C\)$$ respectively and are such that $$\(A D=A E=k a\)$$, where $$\(k< 1\)$$. The line $$\(D E\)$$ divides the sector into two regions which are equal in area. For the general case in which angle $$\(B A C=\theta\)$$ radians, where $$\(0 1\)$$. Find the set of possible values of $$\(k\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_m21_qp_12 Year：2021 Question No：10(b)
Answer:
\( 2 \times \frac{1}{2}(k a)^{2} \sin \theta=\frac{1}{2} a^{2} \theta \)
\( k^{2}=\frac{\theta}{2 \sin \theta} \)
\( k^{2}>\frac{1}{2} \)leading to \( \frac{1}{\sqrt{2}} \)<k<1
\( k^{2}=\frac{\theta}{2 \sin \theta} \)
\( k^{2}>\frac{1}{2} \)leading to \( \frac{1}{\sqrt{2}} \)<k<1
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
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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