Ellipse passing through two given points

Question

Find the equation of ellipse passing through the points (3,4) and(5,1). Are there finite no of ellipse passing through the points?

Answer 1

Five points define a conic, you don't have enough information to define an ellipse unless it is restricted to standard form:

$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$

Substituting your points $A(3,4)$ and $B(5,1)$ gives $15x^2+16y^2=391$

Answer 2

Hint:

The equation of an ellipse referred to its symmetry axis contains two parameters ( do you know ?), so if we know the coordinates of two points we can write two equations and find the two parameters.