Mark the correct alternative in the following:
The equation of the plane through the intersection of the planes ax + by + cz + d = 0 and lx + my + nz + p = 0 and parallel to the line y = 0, z = 0
The equation of the plane through the intersection of
the planes ax + by + cz + d=0 and lx + my + nz + p=0 is given as,
(ax + by + cz + d) + λ(lx + my + nz + p)=0
[where λ is a scalar]
x(a + lλ) + y(b + mλ) + z(c + nλ) + d + pλ=0
Given, that the required plane is parallel to the line y=0, z=0 i.e. x - axis so, we should have,
1(a + lλ) + 0(b + mλ) + 0(c + nλ)=0
a + lλ=0
Substituting the value of λ we get,
(alx + bly + clz + dl) - a(lx + my + nz + p)=0
alx + bly + clz + dl - alx + amy + anz + ap=0
bly + clz + dl - amy - anz - ap=0
(bl - an)y + (cl - an)z + dl - ap=0
Therefore, the equation of the required plane is
(bl–am)y + (cl–an)z + dl–ap=0