Find-the-equation-of-the-plane-passing-through-the-points-1-0-0-and-0-1-0-and-makes-an-angle-of-pi-4-with-the-plane-x-y-3-

Question Number 55475 by Knight last updated on 25/Feb/19

Find the equation of the plane

passing through the points

(1, 0, 0) and (0, 1, 0) and makes

an angle of \frac{π}{4} with the plane

x + y = 3

Answered by tanmay.chaudhury50@gmail.com last updated on 25/Feb/19

l e t e q n o f p l a n e a (x - 1) + b (y - 0) + c (z - 0) = 0

w h i c h p a s s e s t h r o u g h (0, 1, 0)

a (0 - 1) + b (1 - 0) + c (0 - 0) = 0

- a + b = 0 [a = b]

a n g l e b e t w e e n a (x - 1) + b (y - 0) + c (z - 0) = 0

a n d x + y + 0 . z = 3 i s

c o s \frac{π}{4} = \frac{a \times 1 + b \times 1 + c \times 0}{\sqrt{a^{2} + b^{2} + c^{2}} \times \sqrt{1^{2} + 1^{2} + 0^{2}}}

n o s

\frac{1}{\sqrt{2}} = \frac{a + b}{\sqrt{a^{2} + b^{2} + c^{2}} \times \sqrt{2}}

1 = \frac{a + a}{\sqrt{a^{2} + a^{2} + c^{2}}}

4 a^{2} = 2 a^{2} + c^{2}

c = \pm a \sqrt{2}

s o t h e e q n o f p l a n e s a r e

a (x - 1) + b (y - 0) + c (z - 0) = 0

a (x - 1) + a (y - 0) + a \sqrt{2} (z - 0) = 0

x - 1 + y + \sqrt{2} z = 0

x + y + \sqrt{2} z = 1

a n d

a (x - 1) + b (y - 0) - a \sqrt{2} (z - 0) = 0

x - 1 + y - \sqrt{2} z = 0

x + y - \sqrt{2} z = 1