Problem 44 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the standard form of the equation of the hyperbola with the given characteristics. Foci: (±10,0)\(;\) asymptotes: \(y=\pm \frac{3}{4} x\)

Step-by-Step Solution

Answer

The standard form of the equation of the the hyperbola is \(x^2/100 - y^2/56.25 = 1\).

Step 1: Find semi-transverse axis (a)

For a hyperbola, the distance between the foci is given by \(2ae\) where \(a\) is the semi-transverse axis and \(e\) is the eccentricity. Since the foci are at \((±10,0)\), the distance between the foci is \(20\). But since \(e\) for a hyperbola is always more than 1, \(2ae > 2a\), and so, \(a ≤ 10\). Thus, \(a = 10\).

Step 2: Determine semi-conjugate axis (b)

The slope of the asymptotes of a hyperbola is ±b/a. Given equations of the asymptotes are \(y = ± \frac{3}{4}x\), so b/a is equal to \( 3/4\). We’ve found a to be \(10\) in the first step, so: \( b/a = 3/4 \rightarrow b = 10* 3/4 = 7.5 \)

Step 3: Write the Standard Form of the Equation

Insert the calculated values of a and b into the standard equation of the hyperbola. Since the hyperbola is opening to the sides due to the location of the foci, the standard form of the equation of the hyperbola is \(x^2/a^2 - y^2/b^2 = 1\). Replacing \( a = 10 \) and \( b = 7.5 \), this becomes \(x^2/10^2 - y^2/7.5^2 = 1\).