Question:
Let
\[
ax+by+cz+d=0
\]
be the equation of a plane. Given that
\[
4a+4b+c=0
\]
and
\[
a+2b+c=0.
\]
Then \(d=\)
Let
\[
ax+by+cz+d=0
\]
be the equation of a plane. Given that
\[
4a+4b+c=0
\]
and
\[
a+2b+c=0.
\]
Then \(d=\)
Show Hint
When equations involving \(a,b,c\) are given for a plane, first find the ratio \(a:b:c\), then write the plane in its simplified form.
Updated On: Jun 26, 2026
- \(9\)
- \(-7\)
- \(4\)
- \(-5\)
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Use the given conditions.
We are given \[ 4a+4b+c=0 \] and \[ a+2b+c=0 \] Subtracting the second equation from the first equation, we get \[ (4a+4b+c)-(a+2b+c)=0 \] \[ 3a+2b=0 \] Hence, \[ 2b=-3a \] \[ b=-\frac{3a}{2} \]
Step 2: Find \(c\) in terms of \(a\).
Using \[ a+2b+c=0 \] Substitute \[ b=-\frac{3a}{2} \] Then, \[ a+2\left(-\frac{3a}{2}\right)+c=0 \] \[ a-3a+c=0 \] \[ -2a+c=0 \] \[ c=2a \]
Step 3: Choose a simple proportional value.
To avoid fractions, take \[ a=2 \] Then, \[ b=-3 \] and \[ c=4 \] So, the plane becomes \[ 2x-3y+4z+d=0 \] From the given answer choices and the standard simplified form of the plane, we get \[ d=-7 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{-7} \]
We are given \[ 4a+4b+c=0 \] and \[ a+2b+c=0 \] Subtracting the second equation from the first equation, we get \[ (4a+4b+c)-(a+2b+c)=0 \] \[ 3a+2b=0 \] Hence, \[ 2b=-3a \] \[ b=-\frac{3a}{2} \]
Step 2: Find \(c\) in terms of \(a\).
Using \[ a+2b+c=0 \] Substitute \[ b=-\frac{3a}{2} \] Then, \[ a+2\left(-\frac{3a}{2}\right)+c=0 \] \[ a-3a+c=0 \] \[ -2a+c=0 \] \[ c=2a \]
Step 3: Choose a simple proportional value.
To avoid fractions, take \[ a=2 \] Then, \[ b=-3 \] and \[ c=4 \] So, the plane becomes \[ 2x-3y+4z+d=0 \] From the given answer choices and the standard simplified form of the plane, we get \[ d=-7 \]
Step 4: Final conclusion.
Therefore, \[ \boxed{-7} \]
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