Ellipses - please help!!! (1 Viewer)

[Sunyata]

1. Find the equation of the tangents to the ellipse x^2/25 + y^2/9 = 1 which are parallel to the diameter y = 2x.

2. The tangent at P on the ellipse x^2/a^2 + y^2/b^2 =1 cuts the x axis at T and the perpendicular PN is drawn to the x-axis. If O is the origin, prove that ON * OT = a^2.

3. Find the equation of the normal l to 9x^2 + 25y^2 = 225 at P(3, 2 2/5). This normal cuts the x-axis at G and N is the foot of the perpendicular drawn from P to the x-axis. Find GN.

4. Find the equation of the tangent at (4,-1) to the ellopse 9x^2 + 25y^2 = 169. Prove that the circle x^2 + y^2 + 28x - 23y = 152 touches the ellipse at this point.

5. FInd the equations of the normal to x^2 + 4y^2 = 100 at P(8,3). If the normal at P meets the major axis in G, and OY is the perpendicular from O to the tangent at P, prove that PG * OY is equal to the square on the minor semi-axis.

 

[Sunyata]

Could someone please please give me hints on how to do these questions??? I've been trying to work them out all day! PLEASE!!!!!

 

[unholybuddha]

your questions require you to find the tangent/normal so basically for these questions you need to know how to implicitly differentiate.
so for example

a) 2x/25 + (2y/9)(dy/dx) = 0
dy/dx = (-9x)/(25y).
to be parallel to y = 2x, the gradient of tangent = 2.
let dy/dx = 2
(-9x)/(25y) = 2
x = -50y/9 sub this into the equation of the ellipse.
you should get y= 9/root109, x = -50/root109
or y= -9/root109, x= 50/root109
after this you should use the point gradient formula to find the equation of the tangents.