Question

A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point :0 

• (1) (1, –4, 1) 
• (2) (1, 4, –1) 
• (3) (2, 4, 1) 
• (4) (2, –4, 1) 

Answer 1

Correct option is (4) (2, –4, 1)

Explaination::

From question, the equations of planes given are:

2x – y + 2z – 4 = 0

X + 2y + 2z – 2 = 0

The equation of the planes bisecting the angles between two given planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is given as:

\(\frac{{{a_1}x + {b_1}y + {c_1}z + {d_1}}}{{\sqrt {\left( {a_1^2 + b_1^2 + c_1^2} \right)} }} = \pm \frac{{{a_2}x + {b_2}y + {c_2}z + {d_2}}}{{\sqrt {\left( {a_2^2 + b_2^2 + c_2^2} \right)} }}\)

Now, for this problem, the equation of the planes bisecting the angles is:

\(\Rightarrow \frac{{2x - y + 2z - 4}}{{\sqrt {{2^2} + {{( - 1)}^2} + {2^2}} }} = \pm \left( {\frac{{x + 2y + 2z - 2}}{{\sqrt {{1^2} + {2^2} + {2^2}} }}} \right)\)

\(\Rightarrow \frac{{2x - y + 2z - 4}}{{\sqrt 9 }} = \pm \left( {\frac{{x + 2y + 2z - 2}}{{\sqrt 9 }}} \right)\)

\(\Rightarrow \frac{{2x - y + 2z - 4}}{3} = \pm \left( {\frac{{x + 2y + 2z - 2}}{3}} \right)\)

⇒ 2x – y + 2z – 4 = ±(x + 2y + 2z – 2)

Case I: take positive sign

⇒ 2x – y + 2z – 4 = x + 2y + 2z - 2

⇒ x – 3y – 2 = 0 ----(1)

Case II: take negative sign

⇒ 2x – y + 2z – 4 = -(x + 2y + 2z - 2)

⇒ 2x – y + 2z – 4 = -x – 2y – 2z + 2

⇒ 3x + y + 4z – 6 = 0 ----(2)

Now, we need to find which point in the given options satisfies the obtained equation.

The point (2, -4, 1) satisfies equation (2)