Find the equation of the plane passing through the line of intersection of the planes r- - -3- -6-0 and r- -3- - -4k- -0 whose perpendicular distance from origin is unity-

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Image of a Point with Respect to a Plane

Find the equa...

Question

Find the equation of the plane passing through the line of intersection of the planes $\to r . (^i + 3^j) - 6 = 0$ and $\to r . (3^i -^j - 4^k) = 0$ whose perpendicular distance from origin is unity.

Open in App

Solution

$\begin{matrix} P e r p e n d i c u l a r d i s tan c e o f p l a n e f r o m t h e o r i g i n P = ∣ ∣ ∣ \frac{d}{\sqrt{a^{2} + b^{2} + c^{2}}} ∣ ∣ ∣ e q u^{n} o f p l a n e p a s sin g t h r o u g h t h e l i n e o f i n t e r sec t i o n o f t w o p l a n e i s (a_{1} x + b_{1} y + c_{1} z + d_{1}) + λ (a_{2} x + b_{2} y + c_{2} z + d_{2}) = 0 L e t t h e e q u a t i o n o f t h e p l a n e p a s sin g t h o r u g h t h e l i n e o f i n t e r sec t i o n o f t h e p l a n e s x + 3 y - 6 = 0 a n d 3 x - y - 4 z = 0 x + 3 y - 6 + λ (3 x - y - 4 z) = 0 \Rightarrow x (1 + 3 λ) + y (3 - λ) + z (0 - 4 λ) + (- 6) = 0 N o w, ∣ ∣ ∣ \frac{d}{\sqrt{a^{2} + b^{2} + c^{2}}} ∣ ∣ ∣ = 1 \Rightarrow ∣ ∣ ∣ ∣ \frac{- 6}{\sqrt{{(1 + 3 λ)}^{2} + {(3 - λ)}^{2} + {(- 4 λ)}^{2}}} ∣ ∣ ∣ ∣ \Rightarrow \sqrt{26 λ^{2} + 10} = 6 \Rightarrow 26 λ^{2} + 10 = 36 \Rightarrow λ^{2} = 1 ∴ λ = \pm 1 p u t t i n g λ = + 1 7 x + 2 y - 4 z - 6 = 0 \Rightarrow 2 x + y - 2 z - 3 = 0 w h e n λ = - 1 \Rightarrow - 2 x + 4 y + 4 z - 6 = 0 x - 2 y - 2 z + 3 = 0 \end{matrix}$

Suggest Corrections

Similar questions

Find the equation of the plane through the intersection of the planes $\to r . (^i + 3^j) - 6 = 0$ and $\to r . (3^i -^j) - 4^k = 0$ , whose perpendicular distance from origin is unity.

Q. The equation of the plane passing through the line of intersection of the planes

\to r \cdot (2^i - 3^j + 4^k) = 1

and

\to r \cdot (^i -^j) + 4 = 0

and perpendicular to the plane

\to r \cdot (2^i -^j -^k) + 4 = 0,

Q. Prove that equation of the plane passing through (1, 2, 3) and perpendicular to planes

\to r . (^i +^j +^k) = 3

and

\to r . (2^i + 3^j + 4^k) = 0

\to r . (^i - 2^j -^k) = 0

Q. Find the Cartesian equation of the plane, passing through the line of intersection of the planes:

\to r . (2^i + 3^j - 4^k) + 5 = 0

and

\to r . (^i - 5^j + 7^k) + 2 = 0

and intersecting

y

-axis at

(0, 3)

Q. Find the equation of the plane passing through the line of intersection of the planes

\to r . (^i +^j +^k) = 1

and

\to r . (2^i + 3^j -^k) + 4 = 0

and parallel to x-axis.

Join BYJU'S Learning Program

Find the equation of the plane passing through the line of intersection of the planes →r.(^i+3^j)−6=0 and →r.(3^i−^j−4^k)=0 whose perpendicular distance from origin is unity.

Find the equation of the plane passing through the line of intersection of the planes $\to r . (^i + 3^j) - 6 = 0$ and $\to r . (3^i -^j - 4^k) = 0$ whose perpendicular distance from origin is unity.