Find the perpendicular line of y = 1/4x + 8 that passes through the point (-2, 5).

Understand the Problem

The question is asking us to find the equation of the line that is perpendicular to the given line y = 1/4x + 8 and passes through the point (-2, 5). To solve this, we first need to determine the slope of the perpendicular line, which is the negative reciprocal of the slope of the original line. Then we can use the point-slope form of a line to find the equation.

Answer

The equation of the line is $ y = -4x - 3 $.

Steps to Solve

Identify the slope of the original line

The given line is in slope-intercept form: $y = \frac{1}{4}x + 8$.

Here, the slope ( m ) of the line is ( \frac{1}{4} ).

Calculate the slope of the perpendicular line

The slope ( m_{\perp} ) of a line that is perpendicular to another line is the negative reciprocal of the original slope.

Thus:

$$ m_{\perp} = -\frac{1}{\left(\frac{1}{4}\right)} = -4 $$

Use the point-slope form of the line

The point-slope form of a line is given by the equation:

$$ y - y_1 = m(x - x_1) $$

Using the slope ( m_{\perp} = -4 ) and the point (-2, 5), where ( x_1 = -2 ) and ( y_1 = 5 ):

$$ y - 5 = -4(x + 2) $$

Simplify the equation

Now, we'll distribute and simplify:

$$ y - 5 = -4x - 8 $$

Adding 5 to both sides gives:

$$ y = -4x - 3 $$

Write the final equation

The equation of the line that is perpendicular to the original line and passes through (-2, 5) is:

$$ y = -4x - 3 $$

The equation of the line is ( y = -4x - 3 ).

More Information

This equation represents a line that has a slope of -4 and intersects the y-axis at -3. The negative slope indicates that the line goes downwards as you move from left to right. The concept of perpendicular slopes is essential in various applications involving geometry and physics.

Tips

Mixing up the slopes: A common mistake is calculating the slope of the perpendicular line incorrectly. Always remember to take the negative reciprocal.
Failing to use the correct point in the point-slope form: Ensure you use the coordinates of the specified point accurately.

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