Ask Singapore Homework?

Upload a photo of a Singapore homework and someone will email you the solution for free.



Question

secondary 4 | E Maths
One Answer Below

Anyone can contribute an answer, even non-tutors.

Answer This Question
Ivey
Ivey

secondary 4 chevron_right E Maths chevron_right Singapore

〘Circles〙〘 Simultaneous equation〙 How to solve these two questions simultaneously to obtain the point (1.22,0.710) (3s.f.) ?

Date Posted: 3 years ago
Views: 435
J
J
3 years ago
Great , I see that you're ising Desmos as a visualisation aid.

I will type out the answer later.
Ivey
Ivey
3 years ago
Thank you!! :D
J
J
3 years ago
Since both expressions on the Left Hand Side equal to 2, we can equate them.


(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
J
J
3 years ago
Now this is very hard to factorise manually.

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)
Ivey
Ivey
3 years ago
I got it!! Thank you so much~!

See 1 Answer

Answered in main comments. If you're doing O levels A Math, such questions involving two circles will not come out (excluded in 4047 and 4049 syllabi)
done {{ upvoteCount }} Upvotes
clear {{ downvoteCount * -1 }} Downvotes
J
J's answer
1024 answers (A Helpful Person)
Ivey
Ivey
3 years ago
Thank you for the clear explanations and really appreciate your effort for typing down all the steps! Thanks for helping me once again!
J
J
3 years ago
Welcome anytime