Answer:
To find the coordinates of point B on AC such that the ratio of AB to BC is 2:1, we can use the section formula:
Let the coordinates of point B be (x, y).
Then, we know that AB is twice as long as BC. Let's call the length of BC "d".
So, the length of AB is 2d.
Using the distance formula, we can find the distances between the points:
AB = √[(x - 2)² + (y + 8)²]
BC = √[(x + 4)² + (y - 7)²]
Since the ratio of AB to BC is 2:1, we have:
AB/BC = 2/1
(√[(x - 2)² + (y + 8)²]) / (√[(x + 4)² + (y - 7)²]) = 2/1
(√[(x - 2)² + (y + 8)²])² / (√[(x + 4)² + (y - 7)²])² = (2/1)²
[(x - 2)² + (y + 8)²] / [(x + 4)² + (y - 7)²] = 4
Multiplying both sides by [(x + 4)² + (y - 7)²], we get:
4[(x + 4)² + (y - 7)²] = (x - 2)² + (y + 8)²
Expanding both sides, we get:
4x² + 16x + 4y² - 56y + 129 = x² - 4x + 4y² + 16y + 68
Simplifying, we get:
3x² + 20x - 3y² - 72y + 61 = 0
To solve for x and y, we need another equation. We know that point B lies on the line AC, so we can use the slope formula to find the equation of the line:
m = (y₂ - y₁) / (x₂ - x₁)
m = (7 - (-8)) / (-4 - 2)
m = 15 / (-6)
m = -2.5
Using point-slope form, we can find the equation of the line:
y - 7 = -2.5(x + 4)
Simplifying, we get:
y = -2.5x - 2
Now we have two equations:
3x² + 20x - 3y² - 72y + 61 = 0
y = -2.5x - 2
We can substitute the second equation into the first equation to get an equation in terms of x:
3x² + 20x - 3(-2.5x - 2)² - 72(-2.5x - 2) + 61 = 0
Simplifying and solving for x, we get:
x = -3
Substituting x = -3 into y = -2.5x - 2, we get:
y = -9.5
Therefore, the coordinates of point B are (-3, -9.5).
Explanation: