6.6 Q-4

Question Statement

Using the result from Problem 3, find the equation of a hyperbola where the foci are located at $(-5, -5)$ and $(5, 5)$, and the vertices are at $(-3\sqrt{3}, -3\sqrt{2})$ and $(3\sqrt{3}, 3\sqrt{2})$.

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Background and Explanation

The equation of a hyperbola is determined by its foci and vertices. The general property of a hyperbola is:

$$|PF| - |PF'| = 2a,$$

where $PF$ and $PF'$ represent the distances of a point $P(x, y)$ on the hyperbola to the two foci $F(-c, -c)$ and $F'(c, c)$, respectively. Using this relationship and the given geometric properties, we can derive the equation of the hyperbola.

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Solution

Step 1: Determine the value of $a$ from the vertices

The vertices of the hyperbola are $(-3\sqrt{3}, -3\sqrt{2})$ and $(3\sqrt{3}, 3\sqrt{2})$. The distance between the vertices is equal to $2a$:

$$2a = \sqrt{(3\sqrt{3} - (-3\sqrt{3}))^2 + (3\sqrt{2} - (-3\sqrt{2}))^2}.$$

Simplifying:

$$2a = \sqrt{(6\sqrt{3})^2 + (6\sqrt{2})^2}.$$ $$2a = \sqrt{72 + 72} = \sqrt{144} = 12.$$

Thus:

$$a = 6.$$

Step 2: Use the hyperbolic relationship to express the equation

The equation of the hyperbola is based on the relationship:

$$\sqrt{(x + 5)^2 + (y + 5)^2} - \sqrt{(x - 5)^2 + (y - 5)^2} = 2a.$$

Substituting $2a = 12$, we have:

$$\sqrt{(x + 5)^2 + (y + 5)^2} - \sqrt{(x - 5)^2 + (y - 5)^2} = 12.$$

Step 3: Square both sides

Squaring eliminates the square root:

$$\left(\sqrt{(x + 5)^2 + (y + 5)^2} - \sqrt{(x - 5)^2 + (y - 5)^2}\right)^2 = 144.$$

Expand both sides:

$$(x + 5)^2 + (y + 5)^2 + (x - 5)^2 + (y - 5)^2 - 2\sqrt{((x + 5)^2 + (y + 5)^2)((x - 5)^2 + (y - 5)^2)} = 144.$$

Simplify:

$$2x^2 + 2y^2 + 100 - 2\sqrt{((x + 5)^2 + (y + 5)^2)((x - 5)^2 + (y - 5)^2)} = 144.$$

Step 4: Isolate the square root and simplify further

Rearranging:

$$\sqrt{((x + 5)^2 + (y + 5)^2)((x - 5)^2 + (y - 5)^2)} = x^2 + y^2 + 50 - 72.$$

Simplify:

$$\sqrt{((x + 5)^2 + (y + 5)^2)((x - 5)^2 + (y - 5)^2)} = x^2 + y^2 - 22.$$

Step 5: Square again

Squaring again gives:

$$((x + 5)^2 + (y + 5)^2)((x - 5)^2 + (y - 5)^2) = (x^2 + y^2 - 22)^2.$$

Expand both sides and simplify (tedious expansion omitted for brevity). The result is:

$$11x^2 + 11y^2 - 50xy + 5041 = 0.$$

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Key Formulas or Methods Used

1. Hyperbolic Property:

$$|PF| - |PF'| = 2a.$$

2. Distance formula: Used to calculate the distance between vertices.
3. Squaring Technique: Applied twice to eliminate square roots and simplify the equation.
4. Final Hyperbolic Equation:

$$11x^2 + 11y^2 - 50xy + 5041 = 0.$$

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Summary of Steps

1. Find $a$: Compute the distance between vertices to find $2a = 12$, thus $a = 6$.
2. Write the hyperbolic relationship: Use the property of hyperbolas involving the foci and the constant difference.
3. Simplify: Square both sides twice to eliminate the square roots and simplify the resulting equation.
4. Final equation: The hyperbola is given by:

$$11x^2 + 11y^2 - 50xy + 5041 = 0.$$