# 8_7

- Author:
- angela

## 8_7

In a right triangle of sides a and b, an ellipse of semiaxes h, k (parallel to the sides) is inscribed. Show that 2(a − h)(b − k) = ab.

## 8_7

## Solution:

I first created sliders for h and k. Then point C_1 as (h,k). I created a perpendicular line to the x-axis through C_1 (line f) and a line through C_1 perpendicular to the y axis. I then created the intersection points of line g and the y-axis (point Y) and line f and the x-axis (point X). I created the points
F( ) and F_2 ( ). I created the ellipse using the Focus points F and F_2 through point Y. Then I created point A on the y-axis. I created the tangent lines to the ellipse through A (lines I and j) and then the intersection points of the tangents and the ellipse (points B and C). Then, I constructed the polygon CBA. I used the text tool and constructed the equation using the objects and as you can see the equation is true.