The tangents of the ellipse with equation x^2/(a^2)+y^2/(b^2)=1 at the points P(acost, bsint) and Q(-asint, bcost) intersect at the point R. (I tried to find R by calculating the equation of the two tangents and then equating them to obtain the x and y coordinates). As t varies, show that R lies on the curve with equation x^2/(a^2)+y^2/(b^2)=2. I then tried to eliminate t to get an equation with just x's and y's but I couldn't eliminate t, so I must have gone wrong. When I tried to get the x-coordinate of R, everything cancelled to 0.
Could someone help please?
Dec 19th 2007, 12:04 PM
Why try to eliminate the parameter? There is a reason why folks got into parametric equations. Sometimes, you can do more things.
Find the general slope.
You have to know where that came from.
Now, write the equations of the tangent lines.
Don't forget to substitute the right values in dy/dx.
Solve system, for x and y.
I get x = a(cos(t)-sin(t)) and y = b(cos(t)+sin(t))
One last thing. Is that point on the new ellipse?
This is a great review problem. There is a ton of mathematics in there. Understand this problem and you will have some good information for the future.
Dec 20th 2007, 12:32 PM
When working with tangents to conic curves, . . there are some formulas that can save a lot of time and work.