Answer:
The standard form of the ellipse is [tex]\frac{x^{2}}{25} +\frac{y^{2}}{169} = 1[/tex]. Coordinates of the foci are [tex]F_{1} (x,y) = (0,-c)[/tex] and [tex]F_{2}(x,y) = (0,c)[/tex], respectively.
Step-by-step explanation:
The equation of a ellipse centered in the origin in standard form is defined by following formula:
[tex]\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1[/tex] (1)
Where [tex]a, b[/tex] are the coefficients of the ellipse.
Now we proceed to transform the equation of the ellipse in general form into standard form by algebraic means:
[tex]169\cdot x^{2}+25\cdot y^{2}-4225 = 0[/tex]
[tex]169\cdot x^{2}+25\cdot y^{2} = 4225[/tex]
[tex]\frac{x^{2}}{25} +\frac{y^{2}}{169} = 1[/tex]
Since [tex]b > a[/tex], then major axis of the ellipse is located in y-axis. The distance between center and focus ([tex]c[/tex]) is calculated by following Pythagorean identity:
[tex]c =\sqrt{b^{2}-a^{2}}[/tex] (2)
[tex]c = 12[/tex]
The location of the foci are represented by [tex]F_{1} (x,y) = (0,-c)[/tex] and [tex]F_{2}(x,y) = (0,c)[/tex]. If we know that [tex]c = 12[/tex], then the location of the foci are, respectively:
[tex]F_{1}(x,y) = (0,-12)[/tex], [tex]F_{2}(x,y) = (0,12)[/tex]
