Please help me to solve this problem.
Find the equation of the locus of a moving point if the sum of the slopes of the line joining it to (1,3) and (4,-2) is 1.
P(x,y)
m1 = (y-3)/(x-1)
m2 = (y+2)/(x-4)
m1+m2 = 1
(y-3)/(x-1) + (y+2)/(x-4) = 1
(y-3)(x-4) + (y+2)(x-1) = (x-4)(x-1)
xy - 3x -4y+ 12 + xy +2x -y -2 = x^2-5x+4
x^2 - 4 x -2 xy + 5 y - 6 = 0
To solve this problem, we need to find the equation of the locus of a moving point, so let's start step by step.
Step 1: Finding the slope between two points
The slope of the line joining two points (x₁, y₁) and (x₂, y₂) can be found using the formula:
slope = (y₂ - y₁) / (x₂ - x₁)
Step 2: Define the coordinates of the moving point
Let's assume the coordinates of the moving point are (x, y).
Step 3: Find the slope between the moving point and (1,3)
Using the formula from Step 1, we can find the slope between the moving point (x, y) and (1, 3):
slope₁ = (y - 3) / (x - 1)
Step 4: Find the slope between the moving point and (4,-2)
Similarly, find the slope between the moving point (x, y) and (4, -2):
slope₂ = (y - (-2)) / (x - 4)
Step 5: Set up the equation based on the given information
The problem states that the sum of the slopes of the lines joining the moving point to (1,3) and (4,-2) is 1. Thus, we can write the equation as:
slope₁ + slope₂ = 1
Step 6: Substitute the slopes from steps 3 and 4 into the equation
Substitute the previously calculated slopes into the equation from step 5:
(y - 3) / (x - 1) + (y + 2) / (x - 4) = 1
Step 7: Simplify the equation
To simplify this equation, we can eliminate the denominators by multiplying everything by (x - 1)(x - 4):
(y - 3)(x - 4) + (y + 2)(x - 1) = (x - 1)(x - 4)
Step 8: Expand and simplify the equation
Expand the equation from Step 7 and simplify it:
yx - 4y - 3x + 12 + yx - y + 2x - 2 = x² - 5x + 4
2yx - 5y - x - 14 = x² - 5x + 4
Step 9: Rearrange the equation
Rearrange the equation to its standard form:
x² - 5x - (2yx - x + 5y + 18) = 0
x² - (2y + 1)x + 5y + 18 = 0
Therefore, the equation of the locus of the moving point is x² - (2y + 1)x + 5y + 18 = 0.
Please note that this is a general form equation and could be further simplified depending on the context or specific requirements of the problem.