4.3 Q-9

Question Statement

Find the equation of the following lines:
a) The horizontal line through the point $(7, -9)$
b) The vertical line through the point $(-5, 3)$
c) The line bisecting the first and third quadrants
d) The line bisecting the second and fourth quadrants

Background and Explanation

To find the equation of a line, we use the general formula for a line passing through a point $(x_1, y_1)$ :
$y - y_1 = m(x - x_1)$
where $m$ is the slope of the line.

Horizontal Line: A horizontal line has a slope of 0 and has the same $y$ -coordinate for all points on it.
Vertical Line: A vertical line has an undefined slope and has the same $x$ -coordinate for all points on it.
Line Bisecting Quadrants: These lines have specific slopes depending on the angle they make with the axes.

Solution

Part (a): The Horizontal Line Through $(7, -9)$

A horizontal line has the same $y$ -coordinate for all points. The equation of this line is simply:
$y = -9$
This is because the line passes through $(7, -9)$ , and the $y$ -coordinate remains constant for all points on the line.

Part (b): The Vertical Line Through $(-5, 3)$

A vertical line has the same $x$ -coordinate for all points. The equation of the vertical line passing through $(-5, 3)$ is:
$x = -5$
This is because the line passes through $(-5, 3)$ , and the $x$ -coordinate remains constant for all points on the line.

Part (c): The Line Bisecting the First and Third Quadrants

The line bisecting the first and third quadrants has a slope of $\tan 45^\circ = 1$ , as the line makes a $45^\circ$ angle with both axes.

The equation of the line is:
$y = x$
This is because the slope $m = 1$ and the line passes through the origin (where $x = y$ ).

Part (d): The Line Bisecting the Second and Fourth Quadrants

The line bisecting the second and fourth quadrants has a slope of $\tan 135^\circ = -1$ , as the line makes a $135^\circ$ angle with the positive $x$ -axis.

The equation of the line is:
$y = -x$
This is because the slope $m = -1$ and the line passes through the origin (where $x = -y$ ).

Key Formulas or Methods Used

Equation of a Line:
$y - y_1 = m(x - x_1)$
where $m$ is the slope and $(x_1, y_1)$ is a point on the line.
Horizontal Line: The equation is $y = \text{constant}$ .
Vertical Line: The equation is $x = \text{constant}$ .
Line Bisecting Quadrants:
- The slope of the line bisecting the first and third quadrants is $1$ , leading to the equation $y = x$ .
- The slope of the line bisecting the second and fourth quadrants is $-1$ , leading to the equation $y = -x$ .

Summary of Steps

For the horizontal line, use the constant $y$ -coordinate: $y = -9$ .
For the vertical line, use the constant $x$ -coordinate: $x = -5$ .
For the line bisecting the first and third quadrants, use the slope $m = 1$ , resulting in $y = x$ .
For the line bisecting the second and fourth quadrants, use the slope $m = -1$ , resulting in $y = -x$ .