Question:
If \(a,b,c\) are the intercepts made by the plane passing through the point \((1,2,3)\) parallel to the plane \(3x+4y-5z=0\) on \(X,Y,Z\)-axes respectively, then \(3a+b+5c=\)
If \(a,b,c\) are the intercepts made by the plane passing through the point \((1,2,3)\) parallel to the plane \(3x+4y-5z=0\) on \(X,Y,Z\)-axes respectively, then \(3a+b+5c=\)
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Planes parallel to each other have the same coefficients of \(x,y,z\). Only the constant term changes.
Updated On: Jun 25, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Write the equation of the required plane.
The required plane is parallel to \[ 3x+4y-5z=0 \] So, its equation will be \[ 3x+4y-5z+d=0 \]
Step 2: Use the point \((1,2,3)\).
Since the plane passes through \((1,2,3)\), we substitute: \[ 3(1)+4(2)-5(3)+d=0 \] \[ 3+8-15+d=0 \] \[ -4+d=0 \] \[ d=4 \] Hence, the plane is \[ 3x+4y-5z+4=0 \]
Step 3: Find intercepts on coordinate axes.
For \(X\)-intercept, put \(y=0,z=0\): \[ 3x+4=0 \] \[ x=-\frac{4}{3} \] So, \[ a=-\frac{4}{3} \] For \(Y\)-intercept, put \(x=0,z=0\): \[ 4y+4=0 \] \[ y=-1 \] So, \[ b=-1 \] For \(Z\)-intercept, put \(x=0,y=0\): \[ -5z+4=0 \] \[ z=\frac{4}{5} \] So, \[ c=\frac{4}{5} \]
Step 4: Find \(3a+b+5c\).
\[ 3a+b+5c = 3\left(-\frac{4}{3}\right)+(-1)+5\left(\frac{4}{5}\right) \] \[ =-4-1+4 \] \[ =-1 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{-1} \]
The required plane is parallel to \[ 3x+4y-5z=0 \] So, its equation will be \[ 3x+4y-5z+d=0 \]
Step 2: Use the point \((1,2,3)\).
Since the plane passes through \((1,2,3)\), we substitute: \[ 3(1)+4(2)-5(3)+d=0 \] \[ 3+8-15+d=0 \] \[ -4+d=0 \] \[ d=4 \] Hence, the plane is \[ 3x+4y-5z+4=0 \]
Step 3: Find intercepts on coordinate axes.
For \(X\)-intercept, put \(y=0,z=0\): \[ 3x+4=0 \] \[ x=-\frac{4}{3} \] So, \[ a=-\frac{4}{3} \] For \(Y\)-intercept, put \(x=0,z=0\): \[ 4y+4=0 \] \[ y=-1 \] So, \[ b=-1 \] For \(Z\)-intercept, put \(x=0,y=0\): \[ -5z+4=0 \] \[ z=\frac{4}{5} \] So, \[ c=\frac{4}{5} \]
Step 4: Find \(3a+b+5c\).
\[ 3a+b+5c = 3\left(-\frac{4}{3}\right)+(-1)+5\left(\frac{4}{5}\right) \] \[ =-4-1+4 \] \[ =-1 \]
Step 5: Final conclusion.
Therefore, \[ \boxed{-1} \]
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