Question

Find the coordinates of the foot of the perpendicular from (3, − 2) on the line
4x + 5y −1 = 0 .

Answer 1

Answer on Question #65756 – Math / Geometry

Find the coordinates of the foot of the perpendicular from $(3, -2)$ on the line $4x + 5y - 1 = 0$.

Solution.

1. Find perpendicular line:

$$\frac{x - 3}{4} = \frac{y + 2}{5};$$ $$5x - 15 = 4y + 8;$$ $$5x - 4y - 23 = 0;$$

2. Calculate set of equations:

$$\left\{ \begin{array}{l} 4x + 5y - 1 = 0; \\ 5x - 4y - 23 = 0; \end{array} \right.$$ $$\left( \begin{array}{ccc} 4 & 5 & 1 \\ 5 & -4 & 23 \end{array} \right);$$ $$\left( \begin{array}{ccc} 1 & 1.25 & 0.25 \\ 5 & -10.25 & 21.75 \end{array} \right);$$ $$\left( \begin{array}{ccc} 1 & 1.25 & 0.25 \\ 0 & 1 & -\frac{87}{41} \end{array} \right);$$ $$\left( \begin{array}{ccc} 1 & 0 & \frac{119}{41} \\ 0 & 1 & -\frac{87}{41} \end{array} \right);$$ $$\left\{ \begin{array}{l} x = \frac{119}{41}; \\ y = -\frac{87}{41}; \end{array} \right.$$

Answer

Coordinates of the foot is $\left(\frac{119}{41}; -\frac{87}{41}\right)$;

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