Question

Show that the equation of the form
Ax^2+Cy^2+F=0 A,C,and F can not equal 0
Where A and C are of the same sign and F is of opposite sign
(a) Is the equation of an ellipse with center (0,0) if A can not equal C
(b) Is the equation of a circle with center (0,0) A=C

Answer 1

To begin lets state the definition of a circle and ellipse in terms of an equation:
Ellipse:
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\]
Circle:
\[x^2+y^2=r\]
a)A not equal to C
\[Ax^2+Cy^2=F\]
\[\frac{A}{F}x^2+\frac{C}{F}y^2=1\]
Which i beleave you sould be able to finish the argument.Since A/F cant equal C/F because A cant equal C let there be new constants which equal to A/F=1/a^2 and C/F=1/b^2.

Answer 2

b) is the same logic but not as involved. Just use the definition of a circle proved.

Answer 3

Thank you veryy much