Find an equation of the plane.
The plane that contains the line $ x = 1 + t , y = 2 - t , z = 4 - 3t $ and is parallel to the plane $ 5x + 2y + z = 1 $
$5 x+2 y+z=13$
Hello. So the question is taken from victor's and geometrical space. And the question is find the equation of plain. That plane contains the line. It means the plane passes to the same point and from which the line passes. And uh it's popular to the plane five X plus two Y plus that is equal to one. Okay So let me right here that uh that 1 2nd mhm equation oh clean passes to slowly contains the line. Yeah X. is equal to one plus d. Bye. Sure is equal to minus T. And that is equal to 4 -3 T. And he's Caroline buddha clean five x plus two. Y Plus they're difficult to one can be who are you Dennis Since both plane passes to the same point as the line passes and the line passes to so that by taking the T value equals. So we get equation of Linus X -1 is equal to Y -2/-1 And is equal to that minus for -2. Which is the equation of line. So since line and plane passes to the same point so the point from which the plane passes to is 1 to 4. So that equation will be five in two x minus four since it is regular to the given line. So that equation will be five in two X minus four Plus two into Y -2. You can also evaluate by taking it a like since it is particular to the plate. So that will be oh five five X plus two. Y plus said plus lambda is equal to zero since the plane passes to 1 to 4. So from here lambda will be equal to five into 1 plus two into 2. Plus that into for the physical too. Sorry The sources that is not five into 1 Plus two into 2 plus On place of that there will be four is equal to minus of lenders. Or from here lambda will be equal to five plus four plus four. That will be equal to minus off there. Or you can say you can take a minus sign here. Okay? So from here and I was able to 13. So let me substitute this value. And the equation where you get five X plus two Y plus debt minus 13 is equal to zero, which eventually gives the equation of plane is five X plus two. Y plus Z is equal to 13 because the plane and that the human line passes to the same point and it is popular to the given place. So I hope this clears your doubt and thank you