To demonstrate the relationship between the two straight lines given by the equations [tex]\(5x + 12y + 13 = 0\)[/tex] and [tex]\(12x - 5y - 18 = 0\)[/tex], we need to analyze their gradients (slopes) and determine if these lines are perpendicular. Let's go through this step-by-step.
### Step 1: Determine the slope of the first line
The equation of the first line is:
[tex]\[ 5x + 12y + 13 = 0 \][/tex]
To find the slope, we need to rewrite this equation in the slope-intercept form [tex]\( y = mx + c \)[/tex], where [tex]\( m \)[/tex] is the slope.
Solve for [tex]\( y \)[/tex]:
[tex]\[ 12y = -5x - 13 \][/tex]
[tex]\[ y = -\frac{5}{12}x - \frac{13}{12} \][/tex]
From this, we can identify the slope [tex]\( m_1 \)[/tex] of the first line:
[tex]\[ m_1 = -\frac{5}{12} \][/tex]
### Step 2: Determine the slope of the second line
The equation of the second line is:
[tex]\[ 12x - 5y - 18 = 0 \][/tex]
Similarly, rewrite this equation in the slope-intercept form [tex]\( y = mx + c \)[/tex].
Solve for [tex]\( y \)[/tex]:
[tex]\[ -5y = -12x - 18 \][/tex]
[tex]\[ y = \frac{12}{5}x + \frac{18}{5} \][/tex]
From this, we can identify the slope [tex]\( m_2 \)[/tex] of the second line:
[tex]\[ m_2 = \frac{12}{5} \][/tex]
### Step 3: Check if the lines are perpendicular
Two lines are perpendicular if the product of their slopes is [tex]\(-1\)[/tex].
Calculate the product of the slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex]:
[tex]\[ m_1 \cdot m_2 = \left(-\frac{5}{12}\right) \cdot \left(\frac{12}{5}\right) = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], we can conclude that the lines given by the equations:
[tex]\[ 5x + 12y + 13 = 0 \][/tex]
and
[tex]\[ 12x - 5y - 18 = 0 \][/tex]
are perpendicular to each other.
