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An equation in standard form for a hyperbola with center (0, 0), vertex (-5, 0), and focus (-6, 0) is given by y²/25 - x²/9 = 1.
An equation can be defined as a mathematical expression which is used to show and indicate that two (2) or more numerical quantities are equal.
Mathematically, the equation of a hyperbola in standard form is given by:
[tex]\frac{(y\;-\;k)^2}{a^2} - \frac{(x\;-\;h)^2}{b^2} = 1[/tex]
Given the following data:
Center (h, k) = (0, 0)
Vertex (h+a, k) = (-5, 0)
Foci, F = (h+c, k) = (-6, 0) and F' = (6, 0)
Also, we can logically deduce that the value of a and c are -5 and -6 respectively.
For the value of b, we would apply Pythagorean's theorem:
c² = a² + b²
b² = c² - a²
b² = (-6)² - (-5)²
b² = 36 - 25
b² = 9.
b = √9
b = 3.
Substituting the given parameters into the equation of a hyperbola in standard form, we have;
[tex]\frac{(y\;-\;k)^2}{a^2} - \frac{(x\;-\;h)^2}{b^2} = 1\\\\\frac{(y\;-\;0)^2}{-5^2} - \frac{(x\;-\;0)^2}{3^2} = 1\\\\\frac{y^2}{-5^2} - \frac{x^2}{3^2} = 1\\\\\frac{y^2}{25} - \frac{x^2}{9} = 1[/tex]
y²/25 - x²/9 = 1.
In conclusion, we can logically deduce that an equation in standard form for a hyperbola with center (0, 0), vertex (-5, 0), and focus (-6, 0) is given by y²/25 - x²/9 = 1.
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