Answer :
Answer:
- parallel line: a(x-h) +b(y-k) = 0, or y-k = m(x -h); same slope
- perpendicular line: b(x-h) -a(y-k) = 0, or y-k = (-1/m)(x -h); opposite reciprocal slope
Step-by-step explanation:
You want the steps to writing the equation of a line parallel or perpendicular to a given line through a given point.
Equations
Given line ax+by=c and point (h, k), you can write equations for the desired lines using the forms:
a(x -h) +b(y -k) = 0 . . . . . . parallel line
b(x -h) -a(y -k) = 0 . . . . . . . perpendicular line
If you eliminate parentheses and collect terms, you will have the general form equation of the desired line(s). If you solve for y, you will have the slope-intercept form of the equation.
To use these forms, it is helpful to start with the equation of the given line in standard form, as shown. (It could also be in general form ax+by-c=0.)
Steps
Using the above solution, the steps are ...
- Write the equation of the given line in standard form or general form.
- Identify the coefficients 'a' and 'b'.
- Fill in the coefficients and given point coordinates in the relevant equation above.
- Rearrange the result to whatever form you need.
You will note that the same coefficients are used on the same variables for a parallel line. This is because parallel lines have the same slope.
The coefficient of the variables are swapped, and one of them is negated in the equation for the perpendicular line. This is because perpendicular lines have opposite reciprocal slope.
Slope-intercept form
When the given line is in slope-intercept form, y = mx +b, the slope is readily identified as m, the coefficient of x. For the parallel line, this slope can be used directly in the point-slope equation for a line through a given point:
y -k = m(x -h) . . . . . . . line with slope m through point (h, k)
This can be rearranged to the slope-intercept form:
y = mx +(k -mh)
The perpendicular line has opposite reciprocal slope. The line perpendicular to the given line with slope m will have slope -1/m. Then the two forms of equation are ...
y -k = -1/m(x -h) . . . . . . . . . point-slope form
y = (-1/m)x +(k +h/m) . . . . . slope-intercept form
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Additional comment
A standard form equation (ax +by = x) has mutually prime integer coefficients, with a positive leading coefficient. When you have written your equation, you may find it necessary to multiply the equation by some value to put it in this form. For example, -2x +4y = -8 must be multiplied by -1/2 to get the standard form equation x -2y = 4.
You need to be aware of the form required by your grader. The wording "the equation" often means slope-intercept form, or the form matching the given equation or the answer box format.
