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
(a) (bl − am) y + (cl − an) z + dl − ap = 0
(b) (am − bl) x + (mc − bn) z + md − bp = 0
(c) (na − cl) x + (bn − cm) y + nd − cp = 0
(d) None of these

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Solution

The equation of the plane passing through the intersection of the planes
ax + by + cz + d = 0
and lx + my + nz + p = 0
will be (ax + by + cz + d) + λ(lx + my + nz + p) = 0

x(a + λl) + y(b + λm) + z(c + λn) + (d + λp)=0 .......(1)

Since the plane is parallel to the line y=0 and z=0
a + λl=0
λ = $\frac{- a}{l}$
putting the value of λ in equation (1), we get
$x (a + (\frac{- a}{l}) l) + y (b + (\frac{- a}{l}) m) + z (c + (\frac{- a}{l}) n) + d + (\frac{- a}{l}) p = 0$

$y (b l - a m) + z (c l - a n) + d l - a p = 0$

Hence, option (a)

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