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

Engineering

Mathematics

Plane and Its Different Forms

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 :

(2, 4, 1)

(1, 4, –1)

(1, –4, 1 )

(2, –4, 1)

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$\frac{2 x - y + 2 z - 4}{\sqrt{2^{2} + {(- 1)}^{2} + 2^{2}}} = \pm (\frac{x + 2 y + 2 z - 2}{\sqrt{1^{2} + 2^{2} + z^{2}}})$ …(i)

Case I : take positive sign

2x – y + 2z = x + 2y + 2z – 2

x – 3y – 2 = 0 …(ii)

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 …(iii)

Option (4) satisfy eqⁿ (iii)

⇒ (2, –4, 1)