Find the coordinates of the point where the line through the points \left(\right. 3 , - 4 , - 5 \left.\right) and \left(\right. 2 , - 3 , 1 \left.\right) crosses the plane 2 x + y + z = 7.

Engineering

Mathematics

Line of Greatest Slope

Question

Find the coordinates of the point where the line through the points

(3, - 4, - 5)

and

(2, - 3, 1)

crosses the plane

2 x + y + z = 7.

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The equation of line passing through the points

(3, - 4, - 5)

and

(2, - 3, 1)

\frac{x, - 3}{2 - 3} = \frac{y + 4}{- 3 + 4} = \frac{z + 5}{1 + 5} \Rightarrow, \frac{x - 3}{- 1} = \frac{y + 4}{1} = \frac{z + 5}{6}

......(i)

Let the line (i) crosses the plane

2 x + y + z = 7

at point

P (α, β, γ)

...(ii)

∵

P lies on line (i), therefore

(α, β, γ)

must satisfy equation (i)

∴ \frac{α - 3}{- 1} = \frac{β + 4}{1} = \frac{γ + 5}{6} = λ (s a y)

α = - λ + 3; β = λ - 4

and

γ = 6 λ - 5

Also

P (α, β, γ)

lie on plane (ii)

∴ 2 α + β + γ = 7

\Rightarrow 2 (- λ + 3) + (λ - 4) + (6 λ - 5) = 7

\Rightarrow - 2 λ + 6 + λ - 4 + 6 λ - 5 = 7

\Rightarrow 5 λ = 10, \Rightarrow λ = 2

Hence, the coordinate of required point

P

(- 2 + 3, 2 - 4, 6 \times 2 - 5) i . e ., (1, - 2, 7)

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