Let A(2,3) be a fixed point. A point P moves such that PA is equal to the distance of P from the y-axis. Find the equation of the locus of P
PA2 = (x-x1)2 + (y-y1)2 +???
= (x-2)2 + (y-3)2 +???
I tried solving it using the distance formula but I'm still confused because I am given only one point A (2,3) coordinate. I searched all blogs on google but couldn't find a solved problem similar to this. Please help!
review the definition of a parabola.
The standard equation is
y^2 = 4px
if the focus is at (p,0) and directrix x = -p
So, you have to shift your axes giving you
(y-3)^2 = 4(x-1)
see
http://www.wolframalpha.com/input/?i=parabola+(y-3)%5E2+%3D+4(x-1)
To find the locus of point P, we need to determine the set of all possible coordinates (x, y) that satisfy the condition that the distance between P and the point A(2, 3) is equal to the distance between P and the y-axis.
Let's assume the coordinates of point P to be (x, y).
We start by finding the distance between P and A(2, 3) using the distance formula:
PA^2 = (x - 2)^2 + (y - 3)^2
Next, we need to find the distance between P and the y-axis. Since the y-axis is a vertical line passing through the origin, the distance between P and the y-axis is simply the absolute value of the x-coordinate of P.
Therefore, the distance between P and the y-axis is |x|.
Since we are given that PA is equal to the distance of P from the y-axis, we can write:
(x - 2)^2 + (y - 3)^2 = |x|^2
Now, we need to simplify this equation. Since we are dealing with absolute value, we need to consider two cases: one for x ≥ 0 and another for x < 0.
Case 1: x ≥ 0
For x ≥ 0, the absolute value of x is equal to x. Therefore, our equation becomes:
(x - 2)^2 + (y - 3)^2 = x^2
Expanding and simplifying this equation, we get:
x^2 - 4x + 4 + y^2 - 6y + 9 = x^2
Simplifying it further, we have:
-4x + 4 - 6y + 9 = 0
-4x - 6y + 13 = 0
Case 2: x < 0
For x < 0, the absolute value of x is equal to -x. Therefore, our equation becomes:
(x - 2)^2 + (y - 3)^2 = (-x)^2
Expanding and simplifying this equation, we get:
x^2 - 4x + 4 + y^2 - 6y + 9 = x^2
Simplifying it further, we have:
-4x + 4 - 6y + 9 = 0
-4x - 6y + 13 = 0
As you can see, the equations in both cases are the same. This means that regardless of the value of x, the locus of P is defined by the equation:
-4x - 6y + 13 = 0
This is the equation of the locus of point P, which represents all the possible coordinates (x, y) that satisfy the given condition.