Another ellipse construction Start with two circles centered at the origin with radii 0<a<b (see figure). Assume the line ℓ though the origin intersects the smaller circle al Q and the larger circle at R. Let P(x, y) have the y-coordinate of Q and the x-coordinate of R. Show that the set of points P(x, y) generated in this way for all lines ℓ through the origin is an ellipse.

Solution 53RE

Step 1:

In this problem we need to show that the set of points P(x, y) generated in this way for all lines ℓ through the origin is an ellipse.

Given :

Two circles with centers at origin and radii a, b where a<b.

The line l through origin intersects the smaller circle at Q and the larger circle at R.

Let P(x,y) is a point such that it has y-coordinate of Q and x-coordinate of R.

We have to show that the set of points P(x, y) generated in this way for all lines ℓ through the origin is an ellipse.