The plane x − y − z = 4 is rotated through 9 0 ∘ about its line of intersection with the plane x + y + 2 z = 4 and the equation in the new position is A x + B y + C z + D = 0 where A , B , C are least

Question

The plane x − y − z = 4 is rotated through 9 0 ∘ about its line of intersection with the plane x + y + 2 z = 4 and the equation in the new position is A x + B y + C z + D = 0 where A , B , C are least positive integers and D < 0 , then

TestHub · Accepted Answer

The new plane passes through the line of intersection: ( x − y − z − 4 ) + α ( x + y + 2 z − 4 ) = 0 , i.e. ( 1 + α ) x + ( α − 1 ) y + ( 2 α − 1 ) z − ( 4 α + 4 ) = 0 . It is perpendicular to x − y − z = 4 , so ( 1 + α ) − ( α − 1 ) − ( 2 α − 1 ) = 0 , giving α = 2 3 ​ . The plane is 5 x + y + 4 z − 20 = 0 . So A + B + C = 10 , D = − 20 , A BC = 20 and A + B + C + D = − 10 .

Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

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