Question

The foot of the perpendicular from the point $(7, 14, 5)$ to the plane $2x + 4y - z = 2$ are

• A. (1, 2, 8)
• B. (3, 2, 8)
• C. (5, 10, 6)
• D. (9, 18, 4)

Answer 1

The Correct Option is A

The correct answer is A:(1,2,8)
We know that the length of the perpendicular from the point \((x_1, y_1, z_1)\) to the plane 
\(ax + by + cz + d = 0\) is 
\(\frac{\left|ax_{1} + by_{1} +cz_{1} +d\right|}{\sqrt{a^{2} + b^{2} +c^{2}}}\) 
and the co-ordinate \(\left(\alpha,\beta,\gamma\right)\) of the foot of the \(\bot\) are given by 
\(\frac{\alpha-x_{1}}{a} = \frac{\beta-y_{1}}{b} = \frac{\gamma-z_{1}}{c}\)
\(= - \left(\frac{ax_{1} + by_{1} +cz_{1} +d}{a^{2}+b^{2} + c^{2}}\right)\) .......(1) 
In the given ques,, \(x_1 = 7, y_1 = 14, z_1 = 5,\)
\(a = 2 b = 4,c = -1, d = -2\) 
By putting these values in (1), we get 
\(\frac{\alpha -7}{2} = \frac{\beta -14}{4} = \frac{\gamma-5}{-1}= - \frac{63}{21}\)
\(\Rightarrow \, \alpha = 1 , \beta = 2\) and \(\gamma = 8\) 
Hence, foot of \(\bot\) is \((1, 2, 8)\)