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Find the coordinates of the foot of the ...

Find the coordinates of the foot of the perpendicular drawn from the point P(1,-2) on the line y = 2x +1. Also, find the image of P in the line.

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Given line is y=2x+1.
Let M be the foot of the perpendicular drawn from P(1,-2) on the given line.

Slope of given line (AB) is 2.
`therefore " Slope of PM" = -(1)/(2)`
Thus, equation of line PM is
`y + 2 = -(1)/(2)(x-1)`
or `x +2y + 3=0 " "(2)`
Solving equations (1) and (2) , we get M(-1,-1), which is the required foot of the perpendicular on given line. Also, if Q is image of P in the line, then M is the midpoint of PQ.
Therefore, coordinates of point Q are (-3,0).
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