Question

What is the equation, in point-slope form, of the line that is perpendicular to the given line and passes through the point (−4, −3)?

A) y + 3 = −4(x + 4)
B) y + 3 = –1/4(x + 4)
C) y + 3 = 1/4(x + 4)
D) y + 3 = 4(x + 4)

Answer 1

1. Point slope form of the equation of a line is the form y-b=m(x-a),

where the 'point' is (a, b) and the 'slope' is m.

Remark: another way to write y-b=m(x-a) is m=
`((y-b))/((x-a))`, so it simply says that on a line containing points (a,b) and (x,y), the ratio of the differences in y to the differences in x is m.

2. Let m' be the slope of a line perpendicular to the line described. We know that m.m'=-1. (This formula is a result of a trigonometric formula which gives us the tan of the angle between 2 intersection lines. In most cases there is no practical way of doing without this formula so you have to take it as a fact. If you want to understand it deeply I can send you some links.)

3. We need to find m'. Check the pic below to first find m. Fix 2 clearly determined points, (-1, 5) and (-1, 1), so that the y and x differences are correct. The red segment is the y-difference, and it is 4 units, the orange segment is 1 unit.


so the slope m is (difference in y)/(difference in x)= 4/1=4

Careful: the line is decreasing so the slope is negative, thus m=-4

4. m.m'=-1, so m'=-1/m=-1/-4=1/4

5. In y-b=m'(x-a) substitute point (a,b)=(-4, -3) and m=1/4:

y+3=1/4(x+4)