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 the value of \( d \) is:

• A. \( 9 \)
• B. \( -7 \)
• C. \( 4 \)
• D. \( -5 \)

Hint:

Plane Equation with Conditions}
Use linear equations to find ratios of variables
Assume a variable (like \( b = 1 \)) to find specific values
Plug back to compute constants like \( d \)

Answer 1

The Correct Option is B

Given: \[ 4a + 4b + c = 0 \quad \text{(1)}
a + 2b + c = 0 \quad \text{(2)} \] Subtract (2) from (1): \[ (4a + 4b + c) - (a + 2b + c) = 0 \Rightarrow 3a + 2b = 0 \Rightarrow a = -\frac{2b}{3} \] Substitute into (2): \[ -\frac{2b}{3} + 2b + c = 0 \Rightarrow \frac{4b}{3} + c = 0 \Rightarrow c = -\frac{4b}{3} \] So \( a = -\frac{2b}{3},\ c = -\frac{4b}{3} \) Now we consider the form of the plane \( ax + by + cz + d = 0 \) Choose a suitable point (for example, assume \( x = 1,\ y = 1,\ z = 1 \)): \[ a(1) + b(1) + c(1) + d = 0 \Rightarrow a + b + c + d = 0 \Rightarrow \left(-\frac{2b}{3} + b - \frac{4b}{3} + d = 0\right) \Rightarrow -\frac{5b}{3} + d = 0 \Rightarrow d = \frac{5b}{3} \] Now let’s choose \( b = -\frac{21}{5} \Rightarrow d = -7 \)