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Intersection of a 45 degree angle and an ellipse 
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#1
Apr414, 09:18 AM

P: 19

If you are looking at the upper right quadrant of an ellipse centered at (0,0), with a=1 and b=.6, and there is a 45 degree line drawn from (1,.6), how would I find the (x,y) coordinate where the line crosses the ellipse? (I have been out of school for a long time, this is not homework).



#2
Apr414, 10:18 AM

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The line has a slope of 1, and goes through the point (1, .6), so it should be fairly easy to write the line's equation. Once you have equations for the ellipse and the line, solve the system of two equations for the point of intersection. 


#3
Apr414, 10:27 AM

P: 19

Unfortunately, my math is extremely rusty/nonexistent. With someone else's help, I have gotten to y=x.4, and 1.36x^2.8x.2=0 but I don't know how to turn those into (x,y) that I can graph. 


#4
Apr414, 10:32 AM

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P: 21,216

Intersection of a 45 degree angle and an ellipse
For an ellipse whose center is at (0, 0), with vertices at (a, 0) and (0, b), the equation is $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ I don't know how you got 1.36x^2.8x.2=0. BTW, put some spaces in your equations to make them easier to read, like so: 1.36x^2  .8x .2 = 0 


#5
Apr414, 10:45 AM

P: 19

He went from: x^2 + ((x^2 − 0.8x + 0.4^2) / 0.6^2) = 1. To: (1 + 0.6^2) x^2 − 0.8x − (0.6^2 − 0.4^2) = 0. combining the line equation with the ellipse equation, using the quadratic equation. And from that, I got: 1.36x^2  .8x  .2 = 0 


#6
Apr414, 10:50 AM

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The line equation is y = x  .4 Replace y in the first equation with x  .4, and you'll get an equation that is equivalent to the first one you show above. This gives you a quadratic equation, which can be solved by the use of the Quadratic Formula. 


#7
Apr414, 10:58 AM

P: 19

I am coming at this from a very low level of mathematical ability. I can follow along, stepbystep, when I see it, but I don't know how to solve things with the quadratic formula and I don't know how that would get turned into (x,y) that I can graph. 


#8
Apr414, 11:51 AM

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P: 21,216

Quadratic formula: http://en.wikipedia.org/wiki/Quadratic_formula
Your quadratic equation is 1.36x^{2}  .8x  .2 = 0. In the quadratic formula, a = 1.36, b = .8, and c = .2. You should get two values for x  one positive and one negative. Since you're looking at the intersection of the line and ellipse in the first quadrant, you want the positive x value. Substitute the value you find in the equation of your line to get the y value. That (x, y) point will be on both the ellipse and the line. 


#9
Apr414, 10:13 PM

P: 19




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