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Question
Find the length of the perpendicular and the co-ordinates of the foot of the perpendicular from (−10, −2) to the line x + y − 2 = 0
Solution
The coordinate of the foot of the perpendicular from the point (x1, y1) on the line ax + by + c = 0 is
`(x - x_1)/"a" = (y - y_1)/"b"`
= `- ("a"x_1 + "b"y_1 + "c")/("a"^2 + "b"^2)`
∴ The coordinate of the foot of the perpendicular from the point (– 10, – 2) on the line x + y – 2 = 0 is
`(x + 10)/1 = (y + 2)/1`
= `- (10 - 2 - 2)/(1^2 + 1^2)`
x + 10 = y + 2 = `14/2`
x + 10 = y + 2 = 7
x + 10 = 7, y + 2 = 7
x = – 3, y = 5
∴ The required foot of the perpendicular is (– 3, 5).
Length of the perpendicular
= `sqrt((- 10 + 3)^2 + (- 2 - 5)^2`
= `sqrt((- 7)^2 + (- 7)^2`
= `sqrt(49 + 49)`
= `sqrt(2 xx 49)`
= `7sqrt(2)` units
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