Find the coordinates of the point where the line through (3,-4,-5) and (2,-3,1) crosses the plane 2x + y + - Maths - Three Dimensional Geometry

Question

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

Shridhar Kaushik · Accepted Answer

Firstly equation of line passing through (3, –4, –5) and (2, –3, 1) is given by–

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

Coordinates of any general point on this line are given by–

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

Let P be the point of intersection of line and the plane.

So, for some value of λ, coordinates of P will be (–λ + 3, λ – 4, 6λ – 5) [∵ it lies on line]

Also this point lies on the plane, so it will satisfy the equation of plane

$2 x + y + z = 7 \Rightarrow 2 (- λ + 3) + (λ - 4) + (6 λ - 5) = 7 \Rightarrow - 2 λ + 6 + λ - 4 + 6 λ - 5 = 7 \Rightarrow 5 λ - 3 = 7 \Rightarrow 5 λ = 10 \Rightarrow λ = 2$

∴ The coordinates of point of intersection are (–2 + 3, 2 – 4, 6 × 2 – 5)

i.e. (1, –2, 7)

82

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