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Lengths Of Perpendiculars

Length of Perpendiculars Online Tutoring / Homework Help
THEOREM:

To find the length of the perpendicular let fall from a given point upon a given straight line.

Length of perpendicular

1) Let the equation of the straight line be

x cos ∝ + y sin ∝ - p = 0            (1)

If p be the perpendicular on it, then

ON = p and ∠ XON = ∝.

Let the give point be P(x1, y1).

Draw a parallel PR to the given line to meet ON produced in R and draw PQ (which is the required perpendicular).

If OR be p1, then the equation to PR will be,

x cos ∝ + y sin ∝ - p1 = 0

Since this passes through the point (x1, y1), we get

x1 cos ∝ + y1 sin ∝ - p1 = 0

or p1 = x1 cos ∝ + y1 sin ∝

But the required perpendicular = PQ = NR = OR - ON = p1 - p

= x1 cos ∝ + y1 sin ∝ - p             (2)

Thus the length of the required perpendicular is obtained by substituting x1 and y1 for x and y in the given equation.

2) If the equation of the straight line is

Ax + By + C = 0               (3)

Dividing the above equation by √(A2 )+ B2 we get,


Now comparing the above equation with eq. (1) we get,


Hence the perpendicular from the point (x1, y1) on equation (3) = x1 cos ∝+ y1 sin ∝ - p

                                                                                         

Corollary:

The length of perpendicular from the origin i.e. from (0, 0) =


Example:

Find the length of the perpendicular drawn from the point (4, 5) upon the straight line 3x + 4y = 10.

Solution:

We know that for the equation of straight line Ax + By + C = 0 the perpendicular from the point (x1, y1) on it =

Here (x1, y1) = (4, 5) and

the equation of the straight line is 3x + 4y = 10 or 3x + 4y - 10 = 0 so A = 3, B = 4 and C = -10

Lengths of Perpendiculars


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