The gradient of the line between a moving point P(x,y) and the point A(5,3) is equal to the gradient of line PB where B has coordinates (2, -1). Find the eqaution of the locus of P

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We are given two fixed points A(5,3) and B(2,-1). The point P(x,y) moves so that the gradient of PA equals the gradient of PB. Find the equation of the locus of P:

One way to check for the collinearity of three points is to show that the gradient between any two of them is the same, so P lies on AB.

The gradient for the line AB is `m=(3-(-1))/(5-2)=4/3` . The equation of AB is `y-3=4/3(x-5)` or `y=4/3x-11/3`

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The equation of the locus of P is `y=4/3x-11/3`

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