Question

Identify the conic section represented by 5x^2-7x+2y^2=10. Explain.

Answer 1

Why don't you use discriminant test?

Answer 2

It looks as if it may be an ellipse. To be sure and to explain why, you will need to write the equation in that of standard form for an ellipse. See that form at the link:
http://tutorial.math.lamar.edu/Classes/Alg/Ellipses.aspx

Answer 3

A quadratic equation of the form \(Ax^2+Bxy+Cy^2 +Dx+Ey+F=0\) is an ellipse if \(B^2-4AC \lt 0 \)

Here we have \( 5x^2-7x+2y^2=10 \)

A = 5
C= 2
B = 0

So,\( 0^2 - 4 (5) (2) \lt 0 \implies \) Ellipse.

Answer 4

There are even short cut for this, but I am not sure if it is applicable in rigorous/analytic mathematics.