Let point A be (2,7) and point B be (-6,-3). What point is on the segment connecting A and B such that the distance from the point to B is 4 times the distance from the point to A?
The point is P .i find PB=4/5AB and the point P is (0.4, 5). is this correct? thanks
(0.4,5)
69420
Yes, that's what I got.
To find the point P that satisfies the given conditions, we can follow these steps:
1. Find the distance between points A and B:
- Using the distance formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
- Calculating the distance between A (2, 7) and B (-6, -3):
dAB = √((-6 - 2)^2 + (-3 - 7)^2) = √((-8)^2 + (-10)^2) = √(64 + 100) = √(164) = 2√41
2. Determine the distance from point P to point A:
- Let the coordinates of point P be (x, y), so the distance between P and A is:
dPA = √((x - 2)^2 + (y - 7)^2)
3. Determine the distance from point P to point B:
- Since the distance from P to B is four times the distance from P to A, we have:
dPB = 4 * dPA = 4 * √((x - 2)^2 + (y - 7)^2)
4. Set up the equation for the given conditions:
- According to the problem, we know that dPB is equal to 4/5 times the distance between A and B, which gives:
dPB = (4/5) * dAB
5. Set up the equation and solve for P:
- Substitute the respective values into the equation:
4 * √((x - 2)^2 + (y - 7)^2) = (4/5) * (2√41)
- Simplify the equation:
√((x - 2)^2 + (y - 7)^2) = (1/5) * (2√41)
√((x - 2)^2 + (y - 7)^2) = (√41) / 5
- Square both sides of the equation to eliminate the square root:
(x - 2)^2 + (y - 7)^2 = (41/25)
x^2 - 4x + 4 + y^2 - 14y + 49 = 41/25
x^2 - 4x + y^2 - 14y = 41/25 - 49
x^2 - 4x + y^2 - 14y = 41/25 - 1225/25
x^2 - 4x + y^2 - 14y = -1184/25
- Rearrange the equation to standard form:
x^2 - 4x + y^2 - 14y + 1184/25 = 0
- Completing the square for the x and y terms:
(x^2 - 4x + 4) + (y^2 -14y + 49) + (1184/25 - 4 - 49) = 0
(x - 2)^2 + (y - 7)^2 + (1184/25 - 4 - 49) = 0
(x - 2)^2 + (y - 7)^2 + (1184/25 - 100/25 - 196/25) = 0
(x - 2)^2 + (y - 7)^2 + (888/25) = 0
- Simplify further:
(x - 2)^2 + (y - 7)^2 = -888/25
Since the equation has a negative value on the right side, it means that no real point (x, y) exists for which the given condition is satisfied.