Problem 1041 Find the equation for the set of... [FREE SOLUTION]

Chapter 32: Problem 1041

Find the equation for the set of points the difference of whose distances from \((5,0)\) and \((-5,0)\) is 6 units.

Short Answer

Expert verified

The equation of the set of points described in the problem can be found by forming two equations - one for each possible value inside the absolute value: \[ \sqrt{{(x-5)}^2 + y^2} - \sqrt{{(x+5)}^2 + y^2} = 6\] and \[ -\left( \sqrt{{(x-5)}^2 + y^2} - \sqrt{{(x+5)}^2 + y^2} \right) = 6 \] Solving and simplifying these equations, we get the equations of both branches of the hyperbola: \[ {(x-5)}^2 + y^2 = {(x+5)}^2 + y^2 + 12\sqrt{{(x-5)}^2 + y^2} + 36 \] and \[ {(x-5)}^2 + y^2 = {(x+5)}^2 + y^2 - 12\sqrt{{(x+5)}^2 + y^2} + 36 \]

Step by step solution

Understand the Given Information

The foci of the hyperbola are given as (5, 0) and (-5, 0) and the difference in distances is given as 6 units. The hyperbola is centered on the origin because the foci are symmetric about the origin.

Forming the Equation

Using the formula for the difference of distances in a hyperbola, we have \(\left|PF_1 - PF_2\right| = 2a\). If we substitute the given distances and foci into the equation, we can form the following equation: \[ \left| \sqrt{{(x-5)}^2 + y^2} - \sqrt{{(x+5)}^2 + y^2} \right| = 6 \] This is the equation for the set of points described in the problem, but in order to make it more manageable, we need to eliminate the absolute value.

Simplifying the Equation

We can get rid of the absolute value by creating two equations, one for each of the possible values inside the absolute value (the negative and the positive). This gives us: \[ \sqrt{{(x-5)}^2 + y^2} - \sqrt{{(x+5)}^2 + y^2} = 6 \] and \[ -\left( \sqrt{{(x-5)}^2 + y^2} - \sqrt{{(x+5)}^2 + y^2} \right) = 6 \] We'll get two hyperbolas, one for each of the equations. To simplify further, we can square both sides of each equation to eliminate the square roots. This will give us the variations of the equation in a standard form.

Solving the Equations

Simplifying and solving the above two equations yields: \[ {(x-5)}^2 + y^2 = {(x+5)}^2 + y^2 + 12\sqrt{{(x-5)}^2 + y^2} + 36 \quad \text{[This yields the equation of one branch of the hyperbola.]} \] and \[ {(x-5)}^2 + y^2 = {(x+5)}^2 + y^2 - 12\sqrt{{(x+5)}^2 + y^2} + 36 \quad \text{[This is the equation of the other branch of the hyperbola.]} \] We now have the equations of both branches of the hyperbola that fulfill the given conditions.

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91影视

Find the equation for the set of points the difference of whose distances from \((5,0)\) and \((-5,0)\) is 6 units.

Short Answer

Step by step solution

Understand the Given Information

Forming the Equation

Simplifying the Equation

Solving the Equations

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