Coordinate Geometry: equations of tangent to ellipse

[snake1998]

Hi, this is a question I have been set to do over the Christmas break:

1. The ellipse E has the following equation:

. . . . .\(\displaystyle \dfrac{x^2}{a^2}\, +\, \dfrac{y^2}{b^2}\, =\, 1\)

Two points, P and Q, on E have the following coordinates:

. . . . .\(\displaystyle \left(a\, \cos(\theta),\, b\, \sin(\theta)\right)\)

. . . . .\(\displaystyle \left(-a\, \sin(\theta),\, b\, \cos(\theta)\right)\)

Show that the equations of the tangents at P and Q are as follows:

. . . . .\(\displaystyle \left(b\, \cos(\theta)\right)\, x\, +\, \left(a\, \sin(\theta)\right)\,y\, =\, ab\)

. . . . .\(\displaystyle \left(-b\, \sin(\theta)\right)\, x\, +\, \left(a\, \cos(\theta)\right)\,y\, =\, ab\)

2. These two tangents meet at point T. Find the coordinates of T. Show that T lies on the ellipse F with the following equation:

. . . . .\(\displaystyle \dfrac{x^2}{2a^2}\, +\, \dfrac{y^2}{2b^2}\, =\, 1\)

The first question I managed to do quite easily: just a matter of differentiation parametrically and finding the tangents. But I have no idea how to tackle the second question. I tried to equate the two tangents I'd found, but I couldn't simply to find the coordinates. I'm pretty sure these are in terms of a's, b's, and trig.

Thanks for help in advance!

 

[stapel]

Hi, this is a question I have been set to do over the Christmas break:

1. The ellipse E has the following equation:

. . . . .\(\displaystyle \dfrac{x^2}{a^2}\, +\, \dfrac{y^2}{b^2}\, =\, 1\)

Two points, P and Q, on E have the following coordinates:

. . . . .\(\displaystyle \left(a\, \cos(\theta),\, b\, \sin(\theta)\right)\)

. . . . .\(\displaystyle \left(-a\, \sin(\theta),\, b\, \cos(\theta)\right)\)

Show that the equations of the tangents at P and Q are as follows:

. . . . .\(\displaystyle \left(b\, \cos(\theta)\right)\, x\, +\, \left(a\, \sin(\theta)\right)\,y\, =\, ab\)

. . . . .\(\displaystyle \left(-b\, \sin(\theta)\right)\, x\, +\, \left(a\, \cos(\theta)\right)\,y\, =\, ab\)

2. These two tangents meet at point T. Find the coordinates of T. Show that T lies on the ellipse F with the following equation:

. . . . .\(\displaystyle \dfrac{x^2}{2a^2}\, +\, \dfrac{y^2}{2b^2}\, =\, 1\)

The first question I managed to do quite easily: just a matter of differentiation parametrically and finding the tangents. But I have no idea how to tackle the second question. I tried to equate the two tangents I'd found, but I couldn't simply to find the coordinates. I'm pretty sure these are in terms of a's, b's, and trig.

Thanks for help in advance!

Are you maybe supposed to express y (for the intersection point T) in terms of x, a, and b?

When you reply, please include a clear listing of what you've tried for part 2. Thank you!