For the ellipse 4x^2 + 25y^2 = 100,
1) center = (0, 0)
2) vertices = (±5, 0)
3) foci = (0, ±√19)
"The foci of the ellipse lie on the major axis of the ellipse and are equidistant from the origin."
"Each endpoint of the major axis is the vertex of the ellipse."
The standard equations of an ellipse are given as, [tex]\bold{\frac{x^{2} }{a^{2} } + \frac{y^{2} }{b^{2} } =1}[/tex].
here, the center of the ellipse is (0, 0)
the vertices of the ellipse are [tex](\pm a,0)[/tex]
the foci of the ellipse are [tex](0, \pm c)[/tex] where, [tex]c^{2}= a^{2}- b^{2}[/tex]
For given example,
The equation of the ellipse is [tex]4x^{2} + 25y{^2} = 100[/tex]
First we write this equation in standard form of equation of the ellipse,
[tex]\Rightarrow 4x^{2} + 25y{^2} = 100\\\\\Rightarrow \frac{4x^{2} + 25y{^2} }{100} =\frac{100}{100}\\\\ \Rightarrow \frac{4x^{2}}{100} +\frac{25^{2}}{100} =1\\\\\Rightarrow \bold{\frac{x^{2}}{25} + \frac{y^{2}}{4} =1}[/tex]
This equation represents the ellipse with center (0, 0).
Comparing above equation with the standard equation of the ellipse,
we have [tex]a^{2}=25,b^{2}=4[/tex]
[tex]\Rightarrow c^{2}=25-4\\\\\Rightarrow c^{2}=19\\\\\Rightarrow c=\pm \sqrt{19}[/tex]
This means, the foci of the given ellipse are (0, ±√19)
Now, we find the vertices of the ellipse.
⇒ a² = 25
⇒ a = ±5
So, the vertices of the ellipse are (±5, 0).
Therefore, for the ellipse 4x^2 + 25y^2 = 100,
1) center = (0, 0)
2) vertices = (±5, 0)
3) foci = (0, ±√19)
Learn more about the ellipse here:
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