# Find the coordinate of the point P where the line through and crosses the plane passing through three points and Also, find the ratio in which P divides the line segment AB.

Given: A plane passing through L(2, 2, 1), M(3, 0, 1) and N(4, -1, 0).

To find: coordinate of the point P where the line through A(3, -4, -5) and B(2, -3, 1) crosses the plane

The equation of plane crossing through these points is

Applying R2 -R2 + R1 and R3 -R3 + R1

Expanding the determinant,

(x – 2)(2 – 0) – (y – 2)(-1 – 0) + (z – 1)(-3 + 4) = 0

2x – 4 + y – 2 + z – 1 = 0

2x + y + z – 7 = 0

Therefore, the equation of plane is 2x + y + z – 7 = 0

The equation of the line passing through the point (a, b, c) and (m, n, o) is given by:

The equation of the line passing through point A(3, -4, -5) and B(2, -3, 1) is given by:

This point P lies on both line and plane

Therefore,

2(k + 2) + (-k – 3) – 6k + 1 – 7 = 0

2k + 4 – k – 3 – 6k + 1 – 7 = 0

– 5k – 5 = 0

5k = – 5

k = – 1

Hence, Intersection Point P(1, -2, 7)

Section formula:

Let P divide the line AB in the ration k:1 and x1 = 3, x2 = 2

So,

k + 1 = 2k + 3

k = -2

Hence the ratio 2:1 and negative sign shows that P is dividing AB externally

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Find the coordinate of the point P where the line through and crosses the plane passing through three points and Also, find the ratio in which P divides the line segment AB.

Mathematics - Board Papers