Question
If PM is the perpendicular from P (2,3) on to the line x+y=3 then the co-ordinates of M are
Answer: Option C
:
C
If(x,y) is the foot of the perpendicular from (x1,y1) to the line ax + by + c = 0 then
x−x1a=y−y1b=−(ax1+by1+c)a2+b2
Here (x1,y1)=(2,3);ax+by+c=x+y−3
∴ x−21=y−31=−((1×2)+(1×3)+(−3))12+12
⇒ x−21=y−31=−((2)+(3)+(−3))2
⇒ x−21=y−31=−22
⇒ x−21=y−31=−1
∴ x - 2 = -1 ; y-3 = -1
⇒ x = -1 + 2 ; y = -1 + 3
⇒ x = 1 ; y = 2
∴ (x,y) = (1,2)
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:
C
If(x,y) is the foot of the perpendicular from (x1,y1) to the line ax + by + c = 0 then
x−x1a=y−y1b=−(ax1+by1+c)a2+b2
Here (x1,y1)=(2,3);ax+by+c=x+y−3
∴ x−21=y−31=−((1×2)+(1×3)+(−3))12+12
⇒ x−21=y−31=−((2)+(3)+(−3))2
⇒ x−21=y−31=−22
⇒ x−21=y−31=−1
∴ x - 2 = -1 ; y-3 = -1
⇒ x = -1 + 2 ; y = -1 + 3
⇒ x = 1 ; y = 2
∴ (x,y) = (1,2)
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