The line $$\(y=2 x+5\)$$ intersects the circle with equation $$\(x^{2}+y^{2}=20\)$$ at $$\(A\)$$ and $$\(B\)$$. Find the coordinates of $$\(A\)$$ and $$\(B\)$$ in surd form and hence find the exact length of the chord $$\(A B\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w21_qp_13 Year：2021 Question No：9(a)
Answer:
\(x^{2}+(2 x+5)^{2}=20 \quad\) leading to \(x^{2}+4 x^{2}+20 x+25=20\)
\((5)\left(x^{2}+4 x+1\right)[=0]\)
\(x=\frac{-4 \pm \sqrt{16-4}}{2}\)
\(A=(-2+\sqrt{3}, 1+2 \sqrt{3})\)
\(B=(-2-\sqrt{3}, 1-2 \sqrt{3})\)
\(A B^{2}=\operatorname{their}\left(x_{2}-x_{1}\right)^{2}+\) their \(\left(y_{2}-y_{1}\right)^{2}\)
\(\left[A B^{2}=48+12\right.\) leading to \(] A B=\sqrt{60}\)
\((5)\left(x^{2}+4 x+1\right)[=0]\)
\(x=\frac{-4 \pm \sqrt{16-4}}{2}\)
\(A=(-2+\sqrt{3}, 1+2 \sqrt{3})\)
\(B=(-2-\sqrt{3}, 1-2 \sqrt{3})\)
\(A B^{2}=\operatorname{their}\left(x_{2}-x_{1}\right)^{2}+\) their \(\left(y_{2}-y_{1}\right)^{2}\)
\(\left[A B^{2}=48+12\right.\) leading to \(] A B=\sqrt{60}\)
Knowledge points:
1.1.3 solve quadratic equations, and quadratic inequalities, in one unknown (By factorising, completing the square and using the formula.)
1.1.4 solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic
1.3.2 interpret and use any of the forms in solving problems (Including calculations of distances, gradients, midpoints, points of intersection and use of the relationship between the gradients of parallel and perpendicular lines.)
Solution:
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