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8

Find the equation of the plane passing through (a, b, c) and parallel to the plane .

The equation of a plane passing through (x_{1},y_{1},z_{1}) and perpendicular to a line with direction ratios A, B, C is

A(x - x_{1}) + B(y - y_{1}) + C(z - z_{1}) = 0

The plane passes through (a,b,c)

So, x_{1} = a, y_{1} = b, z_{1} = c

Since both planes are parallel to each other, their normal will be parallel

∴ Direction ratios of normal

= Direction ratios of normal of

Direction ratios of normal = (1,1,1)

∴ A = 1, B =1, C = 1

Thus,

Equation of plane in cartesian form is

A(x - x_{1}) + B(y - y_{1}) + C(z - z_{1}) = 0

⇒ 1(x - a) + 1(y - b) + 1(z - c) = 0

⇒ x + y + z - (a + b + c) = 0

Thus, x + y + z = a + b + c is the required equation of plane.

8

6

If the lines and are perpendicular, find the value of k.