Let the plane ax + by + cz = d pass through (2, 3, –5) and is perpendicular to the planes 2x + y – 5z = 10 and 3x + 5y – 7z = 12. If a, b, c, d are integers d > 0 and gcd (|a|, |b|, |c|, d) = 1, then the value of a + 7b + c + 20d is equal to :
18
20
24
22
The Correct Option is D
Solution and Explanation
The correct answer is (D) : 22
Equation of plane through point (2, 3, –5) and perpendicular to planes 2x + y – 5z = 10 and 3x + 5y – 7z = 12 is
\(\begin{vmatrix} x-2 & y-3 & z+5 \\ 2 & 1 & -5 \\ 3 & 5 & -7 \\ \end{vmatrix} = 0\)
∴ Equation of plane is (x – 2) (– 7 + 25) – (y – 3) (– 14 + 15) + (z + 5) · 7 = 0
∴ 18x – y + 7z + 2 = 0
⇒ 18x – y + 7z = – 2
∴ – 18x + y – 7z = 2
On comparing with ax + by + cz = d where d > 0 is
a = – 18, b = 1, c = – 7, d = 2
Therefore a + 7b + c + 20d = 22
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Concepts Used:
Three Dimensional Geometry
Mathematically, Geometry is one of the most important topics. The concepts of Geometry are derived w.r.t. the planes. So, Geometry is divided into three major categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line:
Consider a line L that is passing through the three-dimensional plane. Now, x,y and z are the axes of the plane and α,β, and γ are the three angles the line makes with these axes. These are commonly known as the direction angles of the plane. So, appropriately, we can say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
