I'm studying about ellipse and its properties. My reference is the following pdf:
http://nebula.deanza.edu/~bloom/math43/ellipse-derivation.pdf
My questions are from the very first page of the pdf-file (questions in image below):
Could someone give me analytical proofs? :)
Thnx for any help =)
Because that is the definition of an ellipse. The following calculation will tell you this.
$$\sqrt{(x-(-c))^2+(y-0)^2}+\sqrt{(x-c)^2+(y-0)^2}=A$$ $$\Rightarrow \sqrt{(x+c)^2+y^2}=A-\sqrt{(x-c)^2+y^2}$$ $$\Rightarrow (x+c)^2+y^2=A^2-2\sqrt{(x-c)^2+y^2}+(x-c)^2+y^2$$ $$\Rightarrow 2\sqrt{(x-c)^2+y^2}=A^2+(x-c)^2+y^2-(x+c)^2-y^2$$ $$\Rightarrow 4(x-c)^2+4y^2=(A^2-4cx)^2$$ $$\Rightarrow (1-4c^2)x^2+2c(A^2-1)x+y^2=\frac{A^2}{4}-c^2.$$