Let the foot of the perpendicular from the point \((1, 2, 4)\) on the line \(\frac{x+2}{4}=\frac{y−1}{2}=\frac{z+1}{3}\) be \(P\), Then the distance of \(P\) from the plane \(3x+4y+12z+23=0\) is
- \(5\)
- \(\frac{50}{13}\)
- \(4\)
- \(\frac{63}{13}\)
The Correct Option is A
Solution and Explanation
\(L:\) \(\frac{x+2}{4}=\frac{y−1}{2}=\frac{z+1}{3}\)
Let
\(P=(4t−2,2t+1,3t−1)\)
\(∵ P\) is the foot of perpendicular of \((1, 2, 4)\)
\(∴ 4(4t – 3) + 2(2t – 1) + 3(3t – 5) = 0\)
\(⇒29t=29⇒t=1\)
\(∴ P = (2, 3, 2)\)
Now, distance of \(P\) from the plane
\(3x + 4y + 12z + 23 = 0\), is
\(\begin{vmatrix}\frac{6+12+24+23}{\sqrt{9+16+144}}\end{vmatrix}=\frac{65}{13}=5\)
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Concepts Used:
Distance of a Point From a Line
The length of the perpendicular drawn from the point to the line is the distance of a point from a line. The shortest difference between a point and a line is the distance between them. To move a point on the line it measures the minimum distance or length required.
To Find the Distance Between two points:
The following steps can be used to calculate the distance between two points using the given coordinates:
- A(m1,n1) and B(m2,n2) are the coordinates of the two given points in the coordinate plane.
- The distance formula for the calculation of the distance between the two points is, d = √(m2 - m1)2 + (n2 - n1)2
- Finally, the given solution will be expressed in proper units.
Note: If the two points are in a 3D plane, we can use the 3D distance formula, d = √(m2 - m1)2 + (n2 - n1)2 + (o2 - o1)2.
Read More: Distance Formula