**Find the equation of the set of all points if the distance of P(x,y)*P*(*x*,*y*) to point A(−2,−8)*A*(−2,−8) is always one-half its distance from point B(1,9)*B*(1,9).**

To find the equation of the set of all points \( P(x, y) \) such that the distance from \( P(x, y) \) to point \( A(-2, -8) \) is always one-half the distance from \( P(x, y) \) to point \( B(1, 9) \), we use the following steps.

### Step 1: Express the distance formula
The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

### Step 2: Write the distance expressions for points \( P(x, y) \), \( A(-2, -8) \), and \( B(1, 9) \)

The distance from \( P(x, y) \) to \( A(-2, -8) \) is:
\[
d_{PA} = \sqrt{(x - (-2))^2 + (y - (-8))^2} = \sqrt{(x + 2)^2 + (y + 8)^2}
\]

The distance from \( P(x, y) \) to \( B(1, 9) \) is:
\[
d_{PB} = \sqrt{(x - 1)^2 + (y - 9)^2}
\]

### Step 3: Set up the relationship
We are given that the distance from \( P(x, y) \) to \( A(-2, -8) \) is one-half the distance from \( P(x, y) \) to \( B(1, 9) \). This gives the equation:
\[
d_{PA} = \frac{1}{2} d_{PB}
\]
Substituting the distance formulas:
\[
\sqrt{(x + 2)^2 + (y + 8)^2} = \frac{1}{2} \sqrt{(x - 1)^2 + (y - 9)^2}
\]

### Step 4: Eliminate the square roots
Square both sides of the equation to eliminate the square roots:
\[
(x + 2)^2 + (y + 8)^2 = \frac{1}{4} \left[ (x - 1)^2 + (y - 9)^2 \right]
\]

### Step 5: Expand both sides
First, expand the left side:
\[
(x + 2)^2 + (y + 8)^2 = x^2 + 4x + 4 + y^2 + 16y + 64
= x^2 + y^2 + 4x + 16y + 68
\]

Now expand the right side:
\[
(x - 1)^2 + (y - 9)^2 = x^2 - 2x + 1 + y^2 - 18y + 81 
= x^2 + y^2 - 2x - 18y + 82
\]
Now multiply by \( \frac{1}{4} \):
\[
\frac{1}{4} \left( x^2 + y^2 - 2x - 18y + 82 \right)
= \frac{1}{4}x^2 + \frac{1}{4}y^2 - \frac{1}{2}x - \frac{9}{2}y + \frac{82}{4}
= \frac{1}{4}x^2 + \frac{1}{4}y^2 - \frac{1}{2}x - \frac{9}{2}y + 20.5
\]

### Step 6: Simplify the equation
Now equate the expanded left and right sides:
\[
x^2 + y^2 + 4x + 16y + 68 = \frac{1}{4}x^2 + \frac{1}{4}y^2 - \frac{1}{2}x - \frac{9}{2}y + 20.5
\]

Multiply both sides by 4 to eliminate the fractions:
\[
4(x^2 + y^2 + 4x + 16y + 68) = x^2 + y^2 - 2x - 18y + 82
\]

This simplifies to:
\[
4x^2 + 4y^2 + 16x + 64y + 272 = x^2 + y^2 - 2x - 18y + 82
\]

### Step 7: Collect like terms
Now subtract \( x^2 \), \( y^2 \), \( -2x \), \( -18y \), and 82 from both sides:
\[
(4x^2 - x^2) + (4y^2 - y^2) + (16x + 2x) + (64y + 18y) + (272 - 82) = 0
\]
\[
3x^2 + 3y^2 + 18x + 82y + 190 = 0
\]

### Step 8: Divide by 3
Finally, divide the entire equation by 3:
\[
x^2 + y^2 + 6x + \frac{82}{3}y + \frac{190}{3} = 0
\]

This is the equation of the set of all points \( P(x, y) \) that satisfy the given condition.

---

Would you like further details on any of the steps? Here are 5 related questions for more exploration:

1. How do you find the equation of a circle using the distance formula?
2. How can you generalize this method for other ratios of distances between points?
3. What are conic sections, and how does this problem relate to them?
4. How would the solution change if the ratio was something other than one-half?
5. What is the geometric significance of the resulting equation?

**Tip**: In problems like these, remember that ratios of distances often lead to conic sections (ellipse, hyperbola).

Find the equation of the set of all points if the distance of P(x, y) to point A(−2, −8) is always one-half its distance from point B(1, 9).

This solution finds the equation of all points where the distance from point A(−2, −8) is half the distance from point B(1, 9). The problem applies the distance formula and leads to a conic section equation. Suitable for Grades 10-12.

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