A straight line AB of length 10 units is free to move with its ends on the axes. Find the locus of a point P on the line at a distance of 3 units from the end on the x axis.
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To find the locus of point P on the line, we can start by considering the position of point A on the x-axis.
Let's assume that point A is located at the point (a, 0) on the x-axis, where "a" is a variable representing the x-coordinate of point A.
Since point A is at a distance of 3 units from the y-axis, we can conclude that point B is located at (-a, 0) on the negative x-axis.
Using the distance formula, we can determine the distance between point A (a, 0) and point P (x, y):
√((x - a)² + y²) = 3
Simplifying the equation, we have:
(x - a)² + y² = 3²
(x - a)² + y² = 9
This equation represents the locus of point P on the line.
In conclusion, the locus of point P is a circle centered at (a, 0) on the x-axis, with a radius of 3 units.
To find the locus of point P on the line AB, we need to determine the set of all possible positions of P as the line AB moves with its ends on the axes.
Let's break the problem down into steps:
Step 1: Understand the Problem
We have a straight line AB of length 10 units, with its ends on the axes. We need to find the locus of a point P on the line, which is always at a distance of 3 units from the end on the x-axis.
Step 2: Determine the Position of P
Given that P is always at a distance of 3 units from the end on the x-axis, we can conclude that P moves along a line parallel to the y-axis.
Step 3: Determine the Range of Possible y-coordinates for P
Since P moves along a line parallel to the y-axis, the y-coordinate of P can be any real number. In other words, the range of possible y-coordinates for P is (-∞, ∞).
Step 4: Determine the x-coordinate of the Point P
To determine the x-coordinate of P, we can consider the fact that AB has a length of 10 units. Since the distance from P to the end on the x-axis is 3 units, we can conclude that the distance from P to the end on the y-axis is 10 - 3 = 7 units.
Since AB is a straight line, the ratio of the rise (change in y) to the run (change in x) is constant. In this case, the slope of AB is -7/3, which means for every unit increase in x-coordinate, the y-coordinate decreases by 7/3.
Let's denote the x-coordinate of P as x. We can then write the equation for the line passing through P as:
y = -(7/3)x + C
where C is a constant representing the y-intercept.
Step 5: Determine the value of C
To find the value of C, we substitute the known point (3, 0) on the line into the equation:
0 = -(7/3)(3) + C
0 = -7 + C
C = 7
Step 6: Final Equation for the Locus of P
Now that we know the value of C, we can write the final equation for the locus of point P:
y = -(7/3)x + 7
This equation represents the line along which point P moves as line AB changes its position while keeping its ends on the axes. This line is parallel to the y-axis and passes through the point (0, 7).
Therefore, the locus of point P is the line y = -(7/3)x + 7.