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Question
Answer the following question:
Find the co-ordinates of the foot of the perpendicular drawn from the point P(−1, 3) the line 3x − 4y − 16 = 0
Solution
Let M be the foot of perpendicular drawn from P(–1, 3) to the line
3x – 4y – 16 = 0 ...(i)
Slope of the line 3x – 4y – 16 = 0 is
`(-3)/(-4) = 3/4`
Since PM ⊥ to line (i),
slope of PM = `(-4)/3`
∴ Equation of PM is
y – 3 = `(-4)/3(x + 1)`
∴ 3(y – 3) = –4(x + 1)
∴ 3y – 9 = – 4x – 4
∴ 4x + 3y – 5 = 0 ...(ii)
The foot of perpendicular i.e., point M, is the point of intersection of equation (i) and (ii).
By (i) x 3 + (ii) x 4, we get
25x = 68
∴ x = `68/25`
Substituting x = `68/25` in (ii), we get
`4(68/25) + 3y - 5` = 0
∴ 3y = `5 - 4(68/25) = (125 - 272)/25 = (-147)/25`
∴ y = `(-49)/25`
∴ The co-ordinates of the foot of perpendicular M are `(68/25, (-49)/25)`.
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