Find the equation of the locus of a moving point such that the slope of the line joining the point to A(1,3) is three times the slope of the line joining the point to B(3,1).
let the point be P(x,y)
slope AP = (y-3)/(x-1)
slope BP = (y-1)/(x-3)
(y-3)/(x-1) = 3(y-1)/(x-3)
xy - 3y - 3x + 9 = 3xy -3y - 3x + 3
2xy = 6
xy = 3
Test for reasonableness :
I know P(12, 1/4) lies on it
slope AP = (1/4 - 3)/(12-1)
= -1/4
slope BP = (1/4 - 1)/(12-3)
= -1/12
and -1/4 = 3(-1/12) , as needed
Oh, I see you're in the mood for a mathematical challenge! Well, let's get to it and find that equation, shall we?
First, let's call the coordinates of our moving point (x, y). We know that the slope of the line joining the point to A(1,3) is three times the slope of the line joining the point to B(3,1).
The slope of the line joining the point to A is given by:
m₁ = (y - 3) / (x - 1)
The slope of the line joining the point to B is given by:
m₂ = (y - 1) / (x - 3)
According to the information given, we have:
m₁ = 3 * m₂
So, substituting the respective expressions for m₁ and m₂, we get:
(y - 3) / (x - 1) = 3 * (y - 1) / (x - 3)
Now, let's simplify things a bit and cross-multiply:
(x - 3)(y - 3) = 3(x - 1)(y - 1)
Expanding both sides of the equation, we get:
xy - 3y - 3x + 9 = 3xy - 3x - 3y + 3
Time to simplify some more and get rid of those xy terms:
xy - 3xy - 3y + 3y - 3x + 3x = 3 - 9
Combining like terms:
-2xy = -6
Finally, let's divide both sides by -2 to isolate xy:
xy = 3
And there you have it! The equation of the locus of the moving point is xy = 3.
To find the locus of a moving point, we need to determine the equation that describes the relationship between the coordinates of that point.
Let's denote the moving point as (x, y).
We are given that the slope of the line joining the point to A(1,3) is three times the slope of the line joining the point to B(3,1).
To find the slope between two points (x1, y1) and (x2, y2), we use the formula:
slope = (y2 - y1) / (x2 - x1)
For the line connecting the moving point to A(1,3), the slope is:
slope1 = (3 - y) / (1 - x)
For the line connecting the moving point to B(3,1), the slope is:
slope2 = (1 - y) / (3 - x)
According to the given condition, the slope1 is three times the slope2:
slope1 = 3 * slope2
Substituting the values of slope1 and slope2, we get:
(3 - y) / (1 - x) = 3 * (1 - y) / (3 - x)
Now, we can cross-multiply and simplify the equation:
(3 - y)(3 - x) = 3(1 - y)(1 - x)
Expanding both sides of the equation:
9 - 3x - 3y + xy = 3 - 3x - 3y + 3xy
Simplifying further:
9 - 3y + xy = 3xy
Re-arranging the terms:
9 = 3xy - xy + 3y
Combining like terms:
9 = 2xy + 3y
To obtain the final equation of the locus, we rearrange the terms to isolate y:
2xy + 3y = 9
3y + 2xy = 9
Dividing both sides by 3, we get:
y + (2/3)xy = 3
Therefore, the equation of the locus of the moving point is:
y + (2/3)xy = 3
To find the equation of the locus, we need to analyze the given condition.
Let's consider a moving point P(x, y) on the coordinate plane. We need to determine the equation of the locus of P such that the slope of the line joining P to A(1, 3) is three times the slope of the line joining P to B(3, 1).
The slope of the line joining two points (x1, y1) and (x2, y2) is given by the formula:
slope = (y2 - y1) / (x2 - x1)
Using this formula, let's calculate the slopes of the lines joining P to A and P to B:
slope of line joining P to A = (y - 3) / (x - 1)
slope of line joining P to B = (y - 1) / (x - 3)
According to the given condition, the slope of the line joining P to A is three times the slope of the line joining P to B:
(y - 3) / (x - 1) = 3 * (y - 1) / (x - 3)
Now, let's simplify the equation:
(x - 3)(y - 3) = 3(x - 1)(y - 1)
Expanding the equation:
xy - 3x - 3y + 9 = 3xy - 3x - 3y + 3
Simplifying further:
xy - 3xy = -6
Combining like terms:
-2xy = -6
Dividing both sides by -2:
xy = 3
Thus, the equation of the locus for the moving point P(x, y) is xy = 3.