Question

how can you show that if a normal line to each point on an ellipse passes through the center of an ellipse,then the ellipse is a circle?.

Answer 1

general elipse equation:
\[\frac{(x-a)^2}{h^2}+\frac{(y-b)^2}{k}=1\]
now differentiate implicitly and solve for y'. This will be the slope of your line. Then impose the condition that it passes trough the point (a,b) and you shoud be able to recognize circle equation

Answer 2

derivative:
\[\frac{2(x-a)}{h^2}+\frac{2(y-b)y'}{k^2}=0\]

Answer 3

solving for y':
\[y'=-\frac{2(x-a)k^2}{h^{2}2(y-b}\]
this is the slope of the line.

Answer 4

Equation of the line with this slope that passes trough the (a,b):
\[y-b=-\frac{2(x-a)k^2}{h^{2}2(y-b)}(x-a)\]
now rearange this

Answer 5

and integrate