P5 coordinate geometry Watch
Find equations of the tangent and normal to the ellipse with equation 4x^2 + 25y^2 = 100 at the point P(5cost, 2sint)
dy/dx = -8x/50y
At P, dy/dx = -2cott/5
The equation of the tangent is:
-2cott/5 = (y-2sint)/(x-5cost)
(-2xcost)/(5sint) + (2cos^2t)/(sint) = y-2sint
The answer should be xcost/5 + ysint/2 = 1
How do they get the equation to look so simple?
Havent attempted the normal yet, but thats pretty easy when you know how to do the tangent.
y = 2sint + 2cos^2(t)/sin(t) - 2xcot(t)/5
Multiply by sin(t);
ysin(t) = 2sin^2(t) + 2cos^2(t) - 2xcos(t)/5
(sin^2 + cos^2 = 1), so
ysint = 2 - 2xcos(t)/5
ysint/2 + xcos(t)/5 = 1