Question

A plane \( E_1 \) makes intercepts \( 1, -3, 4 \) on the coordinate axes. The equation of a plane parallel to \( E_1 \) and passing through \( (2,6,-8) \) is

• A. \( \frac{x}{2} - \frac{y}{3} + \frac{z}{4} + 3 = 0 \)
• B. \( \frac{x}{1} - \frac{y}{3} + \frac{z}{4} + 12 = 0 \)
• C. \( \frac{x}{1} - \frac{y}{3} + \frac{z}{4} + 2 = 0 \)
• D. \( \frac{x}{3} - \frac{y}{6} + \frac{z}{2} + \frac{13}{3} = 0 \)

Hint:

Parallel planes always have the same direction ratios (coefficients of \(x,y,z\)).

Answer 1

The Correct Option is C

Step 1: Write the equation of plane \( E_1 \).
Using intercept form, \[ \frac{x}{1} + \frac{y}{-3} + \frac{z}{4} = 1 \] \[ \Rightarrow \frac{x}{1} - \frac{y}{3} + \frac{z}{4} - 1 = 0 \]

Step 2: Equation of a parallel plane.
A plane parallel to \( E_1 \) has the same coefficients: \[ \frac{x}{1} - \frac{y}{3} + \frac{z}{4} + d = 0 \]

Step 3: Substitute the given point \( (2,6,-8) \).
\[ 2 - 2 - 2 + d = 0 \Rightarrow d = 2 \]

Step 4: Final equation.
\[ \boxed{\frac{x}{1} - \frac{y}{3} + \frac{z}{4} + 2 = 0} \]