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Answer :
The foci of the equation y^2 - 25x^2 = 100 are at (±5√3, 0).[tex](a^2 - b^2).[/tex] The foci of the equation y²-25x²=100 are located along the x-axis.
To find the foci of the given equation, we can use the standard form of the equation of an ellipse:
(x - h)²/a² + (y - k)²/b² = 1
Where (h, k) is the center of the ellipse, and a and b are the semi-major and semi-minor axes, respectively.
Step 1: Identify the coefficients in the given equation.
y² - 25x² = 100
Comparing the given equation to the standard form, we can see that:
a = 5 (since [tex]25x^2 = (5x)^2[/tex]
b = 10 (since y^2 = 100 implies b = 10)
The foci of an ellipse are located at (±c, 0), where c = √[tex](a^2 - b^2).[/tex]
Plugging in the values, we get:
c = √[tex](5^2 - 10^2)[/tex]
= √(25 - 100)
= √(-75)
= ±5√3
Therefore, the foci of the equation y^2 - 25x^2 = 100 are at (±5√3, 0).[tex](a^2 - b^2).[/tex]
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