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Equation of The Plane Bisecting the Angle Between Two Planes - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

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Equation of The Plane Bisecting the Angle Between Two Planes is considered one of the most asked concept.
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Equation of The Plane Bisecting the Angle Between Two Planes

Cartesian Form

Equation of the planes bisecting the angle between the planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0 is

$\mathbf{\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}$

Proof:

Given planes are

$\\ \mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}{a_{1} x+b_{1} y+c_{1} z+d_{1}=0} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\text{(i)}\\\mathrm{and\;\;\;\;\;\;\;\;\;\;\;\;\;} {a_{2} x+b_{2} y+c_{2} z+d_{2}=0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\text{(ii)}$

Let P(x, y, z) be a point on the plane bisecting the angle between planes (i) and (ii).

Let PL and PM be the length of perpendiculars from P to planes (i) and (ii).

$\\\therefore \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;PL=PM\\\\\Rightarrow \;\;\;\;\;\;\;\;\;\;\left|\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}\right|=\left|\frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}\right|\\\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\pm \frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}$

This is equation of planes bisecting the angles between the planes (i) and (ii).

Vector Form

Equation of the planes bisecting the angle between the planes $\vec{\mathbf r}\cdot\vec{\mathbf n}_1=d_1 \text{ and } \vec{\mathbf r}\cdot\vec{\mathbf n}_2=d_2$ is

$\left | \frac{\vec{\mathbf r}\cdot\vec{\mathbf n}_1-d_1}{\vec{\mathbf n}_1} \right |=\left | \frac{\vec{\mathbf r}\cdot\vec{\mathbf n}_2-d_2}{\vec{\mathbf n}_2} \right |$

Bisector of the Angle between the Two Planes Containing the Origin

Let the equation of the two planes be
$\\ \mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}{a_{1} x+b_{1} y+c_{1} z+d_{1}=0} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\text{(i)}\\\mathrm{and\;\;\;\;\;\;\;\;\;\;\;\;\;} {a_{2} x+b_{2} y+c_{2} z+d_{2}=0}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\ldots\text{(ii)}$

where d₁ and d₂ are positive.

Then the equation of the bisector of the angle between the planes (i) and (ii) containing the origin is

$\mathbf{\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}= \frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}$

Bisector of the Acute and Obtuse Angle between Two Planes

Let the equation of the two planes be

If a₁a₂ + b₁b₂ + c₁c₂ > 0, then the equation of the bisector of the obtuse angle is,

$\mathbf{\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=\frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}$

If a₁a₂ + b₁b₂ + c₁c₂ < 0, then the equation of the bisector of the obtuse angle is,

$\mathbf{\frac{a_{1} x+b_{1} y+c_{1} z+d_{1}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}}=- \frac{a_{2} x+b_{2} y+c_{2} z+d_{2}}{\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}}$