i) The slope of the line that passes through the points A(2, -1) and B(4, 3) is? ii) The equation of the line passing through points A(2, 1) and B(1, 2) is? iii) The distance from... i) The slope of the line that passes through the points A(2, -1) and B(4, 3) is? ii) The equation of the line passing through points A(2, 1) and B(1, 2) is? iii) The distance from the origin to the line 3x + 4y + 15 = 0 is? iv) The slope of the line 2x + 3y - 6 = 0 is?
Understand the Problem
The question contains multiple-choice problems related to the slope and equations of lines in coordinate geometry, as well as distance calculations from the origin. It asks to find slopes, equations of lines, and distances using provided points and line equations.
Answer
1. i) 2 ii) $x + y - 3 = 0$ iii) 3 iv) $-\frac{2}{3}$
Answer for screen readers
- i) 2 ii) A) $x + y - 3 = 0$ iii) 3 iv) B) $-\frac{2}{3}$
Steps to Solve
- Calculate the slope between points A(2, -1) and B(4, 3)
The formula for the slope ($m$) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Using the points A(2, -1) and B(4, 3):
$$ m = \frac{3 - (-1)}{4 - 2} = \frac{3 + 1}{2} = \frac{4}{2} = 2 $$
- Find the equation of the line passing through points A(2, 1) and B(1, 2)
First, calculate the slope using the same formula:
$$ m = \frac{2 - 1}{1 - 2} = \frac{1}{-1} = -1 $$
Now use the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, using point A(2, 1):
$$ y - 1 = -1(x - 2) $$
Distributing gives:
$$ y - 1 = -x + 2 $$
Rearranging gives:
$$ x + y - 3 = 0 $$
- Calculate the distance from the origin to the line 3x + 4y + 15 = 0
The distance ($d$) from a point $(x_0, y_0)$ to the line Ax + By + C = 0 is given by:
$$ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} $$
For the line 3x + 4y + 15 = 0, A = 3, B = 4, C = 15, and using the origin (0, 0):
$$ d = \frac{|3(0) + 4(0) + 15|}{\sqrt{3^2 + 4^2}} = \frac{|15|}{\sqrt{9 + 16}} = \frac{15}{5} = 3 $$
- Determine the slope of the line 2x + 3y - 6 = 0
To find the slope, rearrange the equation into slope-intercept form ($y = mx + b$):
$$ 3y = -2x + 6 $$
Dividing by 3 gives:
$$ y = -\frac{2}{3}x + 2 $$
Thus, the slope ($m$) is $-\frac{2}{3}$.
- i) 2 ii) A) $x + y - 3 = 0$ iii) 3 iv) B) $-\frac{2}{3}$
More Information
In coordinate geometry, the slope indicates the steepness of a line, while the equation of the line can define its position relative to the axes. The distance from a point to a line uses the coefficients of the line equation, showing how these concepts are intertwined.
Tips
- Confusing the order of points when calculating slope can lead to incorrect values. Always ensure the points are in the correct order: $(x_1, y_1)$ and $(x_2, y_2)$.
- Misapplying the distance formula; remember to always put the coordinates in the correct places (A, B, C) for the line equation.
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