Question

which point on the x - axis lies on the line that passes through point p and is perpendicular to line mn? \\(\bigcirc\\) (0, 1) \\(\bigcirc\\) (0, 4) \\(\bigcirc\\) (1, 0) \\(\bigcirc\\) (4, 0)

Explanation:

Step1: Find slope of line MN

First, identify coordinates: $M(-4, 0)$, $N(4, 2)$.
Slope formula: $m_{MN}=\frac{y_2-y_1}{x_2-x_1}$
$m_{MN}=\frac{2-0}{4-(-4)}=\frac{2}{8}=\frac{1}{4}$

Step2: Find slope of perpendicular line

Perpendicular slope is negative reciprocal:
$m_{\perp}=-4$

Step3: Write line equation through P

Point $P(2, -4)$. Use point-slope form $y-y_1=m(x-x_1)$
$y-(-4)=-4(x-2)$
Simplify: $y+4=-4x+8$ → $y=-4x+4$

Step4: Find x-axis intersection

On x-axis, $y=0$. Substitute into equation:
$0=-4x+4$
Solve for $x$: $4x=4$ → $x=1$
The point is $(1,0)$

Answer:

(1, 0)