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The coordinates of the foot of the perpe...

The coordinates of the foot of the perpendicular from (2,3) to the line `3x+4y - 6 = 0 ` are

A

`(-14/25,-27/25)`

B

`(14/15,-17/25)`

C

`(-14/25,17/25)`

D

`(14/25,27/25)`

Text Solution

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The correct Answer is:
To find the coordinates of the foot of the perpendicular from the point (2, 3) to the line given by the equation \(3x + 4y - 6 = 0\), we can follow these steps: ### Step 1: Identify the line equation and point The line equation is given as: \[ 3x + 4y - 6 = 0 \] We can rewrite it in slope-intercept form \(y = mx + c\): \[ 4y = -3x + 6 \implies y = -\frac{3}{4}x + \frac{3}{2} \] The slope \(m\) of the line is \(-\frac{3}{4}\). ### Step 2: Find the slope of the perpendicular line The slope of the line perpendicular to this line is the negative reciprocal of \(-\frac{3}{4}\): \[ m_{\perpendicular} = \frac{4}{3} \] ### Step 3: Write the equation of the perpendicular line Using the point-slope form of the line equation, the equation of the line passing through the point (2, 3) with slope \(\frac{4}{3}\) is: \[ y - 3 = \frac{4}{3}(x - 2) \] Expanding this: \[ y - 3 = \frac{4}{3}x - \frac{8}{3} \] \[ y = \frac{4}{3}x - \frac{8}{3} + 3 \] \[ y = \frac{4}{3}x + \frac{1}{3} \] ### Step 4: Set the equations equal to find the intersection point Now, we need to find the intersection of the line \(3x + 4y - 6 = 0\) and the perpendicular line \(y = \frac{4}{3}x + \frac{1}{3}\). Substituting \(y\) from the perpendicular line into the line equation: \[ 3x + 4\left(\frac{4}{3}x + \frac{1}{3}\right) - 6 = 0 \] Expanding: \[ 3x + \frac{16}{3}x + \frac{4}{3} - 6 = 0 \] Multiplying through by 3 to eliminate the fraction: \[ 9x + 16x + 4 - 18 = 0 \] \[ 25x - 14 = 0 \implies 25x = 14 \implies x = \frac{14}{25} \] ### Step 5: Substitute back to find \(y\) Now substitute \(x = \frac{14}{25}\) back into the equation of the perpendicular line to find \(y\): \[ y = \frac{4}{3}\left(\frac{14}{25}\right) + \frac{1}{3} \] Calculating: \[ y = \frac{56}{75} + \frac{25}{75} = \frac{81}{75} = \frac{27}{25} \] ### Step 6: Conclusion Thus, the coordinates of the foot of the perpendicular from the point (2, 3) to the line \(3x + 4y - 6 = 0\) are: \[ \left(\frac{14}{25}, \frac{27}{25}\right) \]
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Knowledge Check

  • The coordinates of the foot of the perpendicular from the point (2,3) on the line x+y-11=0 are (i) (-6,5) (ii) (5,6) (iii) (-5,6) (iv) (6,5)

    A
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    B
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    C
    (-5,6)
    D
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