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Question

The lines xa+dαδ=yaα=zadα+δ and xb+cβr=ybβ=zbcβ+r are coplanar and then equation to the plane in which they lie, is?

A
x+y+z=0
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B
xy+z=0
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C
x2y+z=0
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D
x+y2z=0
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Solution

The correct option is B x2y+z=0
The lines will be coplanar, if
∣ ∣adb+caba+dbcα+δαα+δβ+rββ+r∣ ∣=0
Add 3rd column to first and it becomes twice the second and hence the determinant is zero, as the two columns are identical. Again, the equation of the plane in which they lie is
∣ ∣xa+dyazadαδαα+δβrββ+r∣ ∣=0
On adding 1st and 3rd columns and subtracting twice the 2nd, we get
∣ ∣x+z2yyazad0αα+δ0ββ+r∣ ∣=0
[α(β+r)β(α+δ)](x+z2y)=0
(x+z2y)=0.

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