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Question
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by ______.
Options
`37/10, (-1)/10`
`(-1)/10, 37/10`
`10/37, -10`
`2/3, -1/3`
Solution
The coordinates of the foot of perpendiculars from the point (2, 3) on the line y = 3x + 4 is given by `(-1)/10, 37/10`.
Explanation:
Given equation is y = 3x + 4 .....(i)
⇒ 3x – y + 4 = 0
Slope = 3
Equation of any line passing through the point (2, 3) is
y – 3 = m(x – 2) .....(ii)
If equation (i) is perpendicular to eq. (ii)
Then m × 3 = – 1 ......`[because m_1 xx m_2 = - 1]`
⇒ m = `- 1/3`
Putting the value of m in equation (ii) we get
y – 3 = `- 1/3(x - 2)`
⇒ 3y – 9 = – x + 2
⇒ x + 3y = 11 .....(iii)
Solving equation (i) and equation (iii) we get
3x – y = – 4
⇒ y = 3x + 4 ......(iv)
Putting the value of y in eq. (iii) we get
x + 3(3x + 4) = 11
⇒ x + 9x + 12 = 11
⇒ 10x = – 1
⇒ x = `(-1)/10`
From equation (iv) we get
y = `3((-1)/10) + 4`
⇒ y = `(-3)/10 + 4`
⇒ y = `37/10`
So the required coordinates are `((-1)/10, 37/10)`.
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