I Fınding the position of a point on the line

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To find the coordinates of point Q on line l_1 that is closest to point P, the perpendicular distance from P to l_1 must be calculated. The equation of the line l_1 is given as ax + by + c = 0, and the coordinates of P are (x_p, y_p). A common method involves using the formula for the projection of point P onto line l_1, resulting in an expression for Q in terms of P and the coefficients a, b, and c. This approach yields a single equation that defines point Q's coordinates. The discussion emphasizes the need for a concise mathematical representation of Q based on the given parameters.
Arman777
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Let us suppose we have a line ##l_1 = ax + by + c = 0## and we have a point P, ##P(x_p, y_p)## that is outside of this line. If we draw a perpendicular line from point ##P## to a point on the ##l_1##. What would be the coordinates of this point ? I know there are many ways to do it. But I am looking for a single equation that can describe this point (##Q##) ?

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Have you considered using one of those methods to solve the problem, which should give you an expression for Q in terms of P and l_1?
 
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