Question

1. find the distance between the two lines l and p:

line l contains points (4,3)(-2,0).
line p contains point (3,10).

Explanation:

Step1: Find slope of Line L

Calculate slope using two points:
$$m_L = \frac{0 - 3}{-2 - 4} = \frac{-3}{-6} = \frac{1}{2}$$

Step2: Write Line L's equation

Use point-slope form with (4,3):
$$y - 3 = \frac{1}{2}(x - 4)$$
Simplify to standard form $Ax + By + C = 0$:
$$x - 2y + 2 = 0$$

Step3: Confirm Line P is parallel

(Note: For distance between two lines, they must be parallel; this implies Line P has the same slope $\frac{1}{2}$. Write its equation using (3,10):
$$y - 10 = \frac{1}{2}(x - 3)$$
Simplify to standard form:
$$x - 2y + 17 = 0$$

Step4: Apply distance formula

Use formula for distance between parallel lines $Ax+By+C_1=0$ and $Ax+By+C_2=0$:
$$d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$$
Substitute $A=1, B=-2, C_1=2, C_2=17$:
$$d = \frac{|2 - 17|}{\sqrt{1^2 + (-2)^2}} = \frac{15}{\sqrt{5}} = 3\sqrt{5}$$

Answer:

$3\sqrt{5}$