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secondary 4 | E Maths
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〘Circles〙〘 Simultaneous equation〙 How to solve these two questions simultaneously to obtain the point (1.22,0.710) (3s.f.) ?
I will type out the answer later.
(x - 2)² + (y - 4 + 2√5)² = x² + y²
x² - 4x + 4 + (y - 4)² + 2(y - 4)(2√5) + (2√5)² = x² + y²
x² - 4x + 4 + y² - 8y + 16 + 4√5 y - 16√5 + 20 = x² + y²
40 - 16√5 + (4√5 - 8)y - 4x = 0
4x = 40 - 16√5 + (4√5 - 8)y
x = 10 - 4√5 + (√5 - 2)y
Next, substitute 10 - 4√5 + (√5 - 2)y into x² + y² = 2
(10 - 4√5 + (√5 - 2)y)² + y² = 2
(10 - 4√5)² + 2(10 - 4√5)(√5 - 2)y + ((√5 - 2)y)² + y² = 2
100 - 80√5 + 80 + (20√5 - 40 - 40 + 16√5)y + (5 - 4√5 + 4)y² + y² = 2
180 - 80√5 + (36√5 - 80)y + (10 - 4√5)y² = 2
178 - 80√5 + (36√5 - 80)y + (10 - 4√5)y² = 0
Divide by 2,
89 - 40√5 + (18√5 - 40)y + (5 - 2√5)y² = 0
So, use the quadratic formula
x = [-b±√(b² - 4ac)] / 2a
89 - 40√5 + (18√5 - 40)y + (5 - 2√5)y² = 0
y = [-(18√5 - 40)±√( (18√5 - 40)² - 4(5 - 2√5)(89 - 40√5)) ] / ( 2(5 - 2√5) )
y = [20 - 9√5 ± √(18√5 - 40)] / (5 - 2√5)
y ≈ 0.70967 or y ≈ -1.1818
y ≈ 0.710 (3s.f) or y ≈ -1.18 (3s.f)
(A few steps were skipped. Alternatively, just use the calculator directly. And compute each value separately)
x ≈ 10 - 4√5 + (√5 - 2)(0.70967) or x ≈ 10 - 4√5 + (√5 - 2)(-1.1818)
x ≈ 1.2233 or x ≈ 0.77674
x = 1.22 or x = 0.777 (3s.f)
So the coordinates of the two intersection points are (1.22, 0.710) and (-1.18, 0.777)
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