If (2,-1,3) is the foot of the perpendicular drawn from the origin to a plane, then the equation of that plane is
If (2,-1,3) is the foot of the perpendicular drawn from the origin to a plane, then the equation of that plane is
2x + y - 3z + 6 = 0
2x - y + 3z -14 = 0
2x - y + 3z - 13 = 0
2z + y + 3z - 10 = 0
The Correct Option is B
Solution and Explanation
To find the equation of the plane given that the point (2, -1, 3) is the foot of the perpendicular from the origin to the plane, we can proceed with the following steps:
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The vector from the origin to the point (2, -1, 3) is the normal vector to the plane. Therefore, the normal vector is \vec{n} = (2, -1, 3).
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The general equation of a plane with normal vector (a, b, c) passing through a point (x_0, y_0, z_0) is given by:
a(x - x_0) + b(y - y_0) + c(z - z_0) = 0
Plugging in the normal vector components a = 2, b = -1, and c = 3, and the point (x_0, y_0, z_0) = (2, -1, 3), we have:
2(x - 2) - (y + 1) + 3(z - 3) = 0
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Expanding and simplifying the equation:
2x - 4 - y - 1 + 3z - 9 = 0
Simplifying further, we get:
2x - y + 3z - 14 = 0
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This matches the given option 2x - y + 3z - 14 = 0, which confirms it as the correct answer.
After following these steps, we conclude that the correct equation of the plane is 2x - y + 3z - 14 = 0.
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Concepts Used:
Exponential and Logarithmic Functions
Logarithmic Functions:
The inverses of exponential functions are the logarithmic functions. The exponential function is y = ax and its inverse is x = ay. The logarithmic function y = logax is derived as the equivalent to the exponential equation x = ay. y = logax only under the following conditions: x = ay, (where, a > 0, and a≠1). In totality, it is called the logarithmic function with base a.
The domain of a logarithmic function is real numbers greater than 0, and the range is real numbers. The graph of y = logax is symmetrical to the graph of y = ax w.r.t. the line y = x. This relationship is true for any of the exponential functions and their inverse.
Exponential Functions:
Exponential functions have the formation as:
f(x)=bx
where,
b = the base
x = the exponent (or power)