Answer:
The equation of the elipse is
[tex]\frac{x^2}{169} + \frac{y^2}{25} = 1[/tex]
Step-by-step explanation:
Equation of an elipse:
The equation of a elipse with centre [tex](x_c,y_c)[/tex] has the following format:
[tex]\frac{(x - x_c)^2}{a^2} + \frac{(y - y_c)^2}{b^2} = 1[/tex]
Centre at (0,0):
Means that [tex](x_c,y_c) = (0,0)[/tex].
So
[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1[/tex]
Vertex at (13,0):
Vertex at 13 means that [tex]a = 13, a^2 = 13^2 = 169[/tex]
Focus at (-12,0):
Means that [tex]c = 12, c^2 = 12^2 = 144[/tex]
We have that:
[tex]a^2 = b^2 + c^2[/tex]
So
[tex]169 = b^2 + 144[/tex]
[tex]b^2 = 25[/tex]
So, the equation of the elipse is
[tex]\frac{x^2}{169} + \frac{y^2}{25} = 1[/tex]
