Question

9 the equation of the hyperbola is given:

\\\frac{(x+4)^2}{86} - \frac{(y-3)^2}{35} = 1\\

what is the coordinates of the foci?
\\x = \underline{\quad\quad\quad}, y = \underline{\quad\quad\quad}\\
\\x = \underline{\quad\quad\quad}, y = \underline{\quad\quad\quad}\\

Explanation:

🆕 New Concept Discovered: Equations and Properties of Hyperbolas
Finding the center and focal distance of a split curve

Step 1: Identify the standard form and orientation

The given equation of the hyperbola is:

$$ \frac{(x+4)^2}{86} - \frac{(y-3)^2}{35} = 1 $$

This matches the standard form of a horizontal hyperbola:

$$ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 $$

From this, we can identify the center \((h, k)\) and the values of \(a^2\) and \(b^2\):

• Center \((h, k) = (-4, 3)\)
• \(a^2 = 86\)
• \(b^2 = 35\)

Step 2: Calculate the focal distance \(c\)

For a hyperbola, the distance \(c\) from the center to each focus is related to \(a\) and \(b\) by the equation:

$$ c^2 = a^2 + b^2 $$

Substitute the values of \(a^2\) and \(b^2\):

$$ c^2 = 86 + 35 $$

$$ c^2 = 121 $$

$$ c = \sqrt{121} = 11 $$

Step 3: Determine the coordinates of the foci

Since the hyperbola is horizontal (the \(x\)-term is positive), the foci lie along the horizontal line passing through the center. We find their coordinates by adding and subtracting the focal distance \(c\) from the \(x\)-coordinate of the center:

$$ (h \pm c, k) $$

Substitute \(h = -4\), \(c = 11\), and \(k = 3\):

• First focus:

$$ x = -4 + 11 = 7, \quad y = 3 $$

• Second focus:

$$ x = -4 - 11 = -15, \quad y = 3 $$

Answer:

• First Focus: \(x = 7\), \(y = 3\)
• Second Focus: \(x = -15\), \(y = 3\)