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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: JackSonSmith on December 25, 2014, 11:07:03 pm

Title: Another circles intercepts question
Post by: JackSonSmith on December 25, 2014, 11:07:03 pm: Find the coordinates of the points of intersection of the curves

x^2 / 4 + y^2 / 9 = 1 and x^2 / 9 + y^2 / 4 = 1
Title: Re: Another circles intercepts question
Post by: Tyleralp1 on December 25, 2014, 11:25:12 pm: Transpose each side for y:

y=sqrt(9-(9x^2)/4)
y=sqrt(4-(4x^2)/9)

Let the sides equal each other and square both sides:
9-(9x^2)/4 = 4-(4x^2)/9

Solve for x.

Sub answer back into equation, to find the respective y coordinate.
Title: Re: Another circles intercepts question
Post by: Phy124 on December 25, 2014, 11:58:20 pm: Handy knowledge - If you have two ellipses of the form $\frac{x^2}{a}+\frac{y^2}{b}=c$ and $\frac{x^2}{b}+\frac{y^2}{a}=c$ , the points of interception of the two are given by $(x,y) = \left(\pm\sqrt{\frac{abc}{a+b}},\pm\sqrt{\frac{abc}{a+b}}\right)$ .

How was this derived you might question? Simply put, one ellipse is the other flipped about the line $y = x$ (or similarly $y = -x$ ) so we know they will intercept somewhere along these lines. Take one of the equations and sub $x$ in for $y$ , I'll use the first one. We have $\frac{x^2}{a}+\frac{x^2}{b} = c \Leftrightarrow (a+b)x^2 = abc \Leftrightarrow x^2= \frac{abc}{a+b} \Rightarrow x = \pm \sqrt{\frac{abc}{a+b}}$ . We know the solutions are along the lines $y = \pm x$ so the solutions for $y$ are the same and you now have your four points. Yay for killing time waiting to check out boxing day sales by doing maths ::)
Title: Re: Another circles intercepts question
Post by: keltingmeith on December 26, 2014, 12:48:06 am: Quote from: Phy124 on December 25, 2014, 11:58:20 pm
Yay for killing time waiting to check out boxing day sales by doing maths ::)

Then doing more maths because they didn't calculate the new price for you, they just told you the discount. ;)
Title: Re: Another circles intercepts question
Post by: lzxnl on December 26, 2014, 12:54:41 am: Quote from: Tyleralp1 on December 25, 2014, 11:25:12 pm
Transpose each side for y:

y=sqrt(9-(9x^2)/4)
y=sqrt(4-(4x^2)/9)

Let the sides equal each other and square both sides:
9-(9x^2)/4 = 4-(4x^2)/9

Solve for x.

Sub answer back into equation, to find the respective y coordinate.

I'd solve for y^2 instead and set them equal to each other.