A circle with centre $$\(C\)$$ has equation $$\((x-8)^{2}+(y-4)^{2}=100\)$$. Two tangents from $$\(T\)$$ to the circle are drawn. Show that the angle between one of the tangents and $$\(C T\)$$ is exactly $$\(45^{\circ}\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_w20_qp_13 Year：2020 Question No：11(b)
Answer:
angle \(=\sin ^{-1}\left(\frac{\text { their } 10}{\text { their } 10 \sqrt{2}}\right)\)
angle \(=\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right.\) or \(\frac{\sqrt{2}}{2}\) or \(\frac{10}{10 \sqrt{2}}\) or \(\left.\frac{10}{\sqrt{200}}\right)=45^{\circ}\)
Alternative method for question 11(b)
\[
\left(10 \sqrt{2}^{2}\right)^{2}=10^{2}+T A^{2}
\]
\[
T A=10 \rightarrow 45^{\circ}
\]
angle \(=\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right.\) or \(\frac{\sqrt{2}}{2}\) or \(\frac{10}{10 \sqrt{2}}\) or \(\left.\frac{10}{\sqrt{200}}\right)=45^{\circ}\)
Alternative method for question 11(b)
\[
\left(10 \sqrt{2}^{2}\right)^{2}=10^{2}+T A^{2}
\]
\[
T A=10 \rightarrow 45^{\circ}
\]
Knowledge points:
1.3.4 use algebraic methods to solve problems involving lines and circles (Including use of elementary geometrical properties of circles, e.g. tangent perpendicular to radius, angle in a semicircle, symmetry.) (Implicit differentiation is not included.)
1.5.2 use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles
Solution:
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