Question

If y intercept of a plane \( (x - y + z - 1) + q(x + y - z - 1) = 0 \) is 3 then q is :

• A. \( 2 \)
• B. \( 1 \)
• C. \( 1/2 \)
• D. None of the above

Hint:

To find the y-intercept of a plane \( Ax + By + Cz + D = 0 \), set \( x = 0 \) and \( z = 0 \). The equation becomes \( By + D = 0 \), so the y-intercept is \( y = -D/B \) (provided \( B \neq 0 \)). In this problem, \( B = q-1 \) and \( D = -(1+q) \), so the y-intercept is \( -(-(1+q))/(q-1) = (1+q)/(q-1) \).

Answer 1

The Correct Option is A

Step 1: Write down the equation of the plane. \[ (x - y + z - 1) + q(x + y - z - 1) = 0 \] Step 2: Rearrange the equation by grouping the terms with x, y, z, and the constant term. \[ x - y + z - 1 + qx + qy - qz - q = 0 \] \[ (1 + q)x + (-1 + q)y + (1 - q)z + (-1 - q) = 0 \] \[ (1 + q)x + (q - 1)y + (1 - q)z = 1 + q \] Step 3: Find the y-intercept. The y-intercept is the point where the plane intersects the y-axis. At this point, the x and z coordinates are zero (\( x = 0, z = 0 \)). Substitute these values into the plane equation. \[ (1 + q)(0) + (q - 1)y + (1 - q)(0) = 1 + q \] \[ (q - 1)y = 1 + q \] Step 4: Solve for y, which represents the y-intercept. \[ y = \frac{1 + q}{q - 1} \] Step 5: Use the given information that the y-intercept is 3. \[ 3 = \frac{1 + q}{q - 1} \] Step 6: Solve the equation for q. \[ 3(q - 1) = 1 + q \] \[ 3q - 3 = 1 + q \] \[ 3q - q = 1 + 3 \] \[ 2q = 4 \] \[ q = \frac{4}{2} \] \[ q = 2 \] Thus, the value of q is 2.