A diameter of a circle $$\(C_{1}\)$$ has end-points at $$\((-3,-5)\)$$ and $$\((7,3)\)$$. The two circles intersect at points $$\(R\)$$ and $$\(S\)$$. Show that the equation of the line $$\(R S\)$$ is $$\(y=-2 x+13\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_m20_qp_12 Year：2020 Question No：12(c)
Answer:
Gradient \(m\) of line joining centres \(=\frac{4}{8} \mathrm{OE}\)
Attempt to find mid-point of line.
Equation of \(R S\) is \(y-1=-2(x-6)\)
\[
y=-2 x+13
\]
Alternative method for question \(12(c)\)
\[
(x-2)^{2}+(y+1)^{2}-41=(x-10)^{2}+(y-3)^{2}-41 \mathrm{OE}
\]
\[
x^{2}-4 x+4+y^{2}+2 y+1=x^{2}-20 x+100+y^{2}-6 y+9 \mathrm{OE}
\]
\[
16 x+8 y=104
\]
\[
y=-2 x+13
\]
Attempt to find mid-point of line.
Equation of \(R S\) is \(y-1=-2(x-6)\)
\[
y=-2 x+13
\]
Alternative method for question \(12(c)\)
\[
(x-2)^{2}+(y+1)^{2}-41=(x-10)^{2}+(y-3)^{2}-41 \mathrm{OE}
\]
\[
x^{2}-4 x+4+y^{2}+2 y+1=x^{2}-20 x+100+y^{2}-6 y+9 \mathrm{OE}
\]
\[
16 x+8 y=104
\]
\[
y=-2 x+13
\]
Knowledge points:
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.)
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.)
Solution:
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