Question

lf the planes 2x - y - 3z - 7 = 0 and 4x - 2y + 5kz + 9 = 0 are parallel, then 5k^2 + 6 =​

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{The planes 2x-y-3z-7=0 and 4x-2y+5kz+9=0 are parallel}$

$\underline{\textsf{To find:}}$

$\textsf{The value of}\;\mathsf{5k^2+6}$

$\underline{\textsf{Solution:}}$

$\mathsf{Concept:}$

$\textsf{If the planes}\;\mathsf{a_1x+b_1y+c_1z+d_1=0\;and\;a_2x+b_2y+c_2z+d_2=0}\;\textsf{are parallel, then}$

$\mathsf{\bf\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}}$

$\textsf{Since the given two planes are parallel, we have}$

$\mathsf{\dfrac{2}{4}=\dfrac{-1}{-2}=\dfrac{-3}{5k}}$

$\mathsf{\dfrac{1}{2}=\dfrac{1}{2}=\dfrac{-3}{5k}}$

$\mathsf{\dfrac{1}{2}=\dfrac{-3}{5k}}$

$\mathsf{5k=-6}$

$\implies\boxed{\mathsf{k=\dfrac{-6}{5}}}$

$\textsf{Now,}$

$\mathsf{5k^2+6}$

$\mathsf{=5\left(\dfrac{-6}{5}\right)^2+6}$

$\mathsf{=5\left(\dfrac{36}{25}\right)+6}$

$\mathsf{=\dfrac{36}{5}+6}$

$\mathsf{=\dfrac{36+30}{5}}$

$\mathsf{=\dfrac{66}{5}}$

$\underline{\textsf{Answer:}}$

$\mathsf{The\;value\;of\5k^2+6\;is\;\dfrac{66}{5}}$

Answer 2

HELLO DEAR,

GIVEN:- Plane 2x - y - 3z - 7 = 0. And plane 4x - 2y + 5kz + 9 = 0

are parallel.

Then to find the value of 5k²+ 6 = ?

SOLUTION:-the two given planes are

2x - y - 3z - 7 = 0. and 4x - 2y + 5kz + 9 = 0

The condition for two planes are parallel is

a1 / a2 = b1/b2 = c1/c2

If we take plane i) 2x - y - 3z = 7

and plane ii) 4x-2y +5kz = -9

then a1 = 2 , b1 = -1 , c1= -3

and a2= 4, b2= -2 , c2 = 5k

So, a1/a2 = b1 / b2 = c1/c2

by putting the value of a1 ,a2,b1,b2,c1,c3

2/4 = -1/-2 =-3/5k

⇒ 2/4=1/2=-3/5k

⇒-3/5k = 1/2

⇒ k = -6/5

And by putting the value of k= -6/5 in

5k²+6.

We get , 5(-6/5)²+6 = 5(36/25) +6

= 36/5 + 6

= 66/5

= 13.2

I HOPE IT'S HELP YOU DEAR,

THANKS.