Question

The equation of plane passing through ( (1, 0, 0) ) and ( (0, 1, 0) ) and making an angle ( 45^\circ ) with the plane ( x + y - 3 = 0 ) is

• A. ( x + y \pm \sqrt{2}z - 1 = 0 )
• B. ( 3x + y \pm \sqrt{3}z - 3 = 0 )
• C. ( x + y \pm \sqrt{3}z - 1 = 0 )
• D. ( 2x + 2y \pm \sqrt{3}z - 2 = 0 )

Hint:

The intercept form of a plane is $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$.

Answer 1

The Correct Option is A

Step 1: General Equation
A plane passing through ((1,0,0)) and ((0,1,0)) has the form (\frac{x}{1} + \frac{y}{1} + \frac{z}{c} = 1) or (x + y + kz - 1 = 0).

Step 2: Use Angle Formula
Angle (\theta) between (x+y+kz-1=0) and (x+y-3=0) is (45^\circ).
(\cos 45^\circ = \frac{|(1)(1) + (1)(1) + (k)(0)|}{\sqrt{1^2+1^2+k^2}\sqrt{1^2+1^2}}).

Step 3: Calculation
(\frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2+k^2}\sqrt{2}} \implies \sqrt{2+k^2} = 2 \implies 2+k^2 = 4 \implies k^2 = 2 \implies k = \pm\sqrt{2}).

Step 4: Conclusion
The equation is (x + y \pm \sqrt{2}z - 1 = 0).
Final Answer: (A)