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Find the coordinates of the point which divides the line segment joining the points (- 2, 3, 5) and (1, - 4, 6) in the ratio 2:3 internally ?
1. \(\left( {\frac{{4}}{{5}},\frac{{1}}{{5}},\frac{{27}}{{5}}} \right)\)
2. \(\left( {\frac{{4}}{{5}},\frac{{1}}{{5}},\frac{{-27}}{{5}}} \right)\)
3. \(\left( {\frac{{4}}{{5}},\frac{{-1}}{{5}},\frac{{-27}}{{5}}} \right)\)
4. \(\left( {\frac{{- 4}}{{5}},\frac{{1}}{{5}},\frac{{27}}{{5}}} \right)\)

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Correct Answer - Option 4 : \(\left( {\frac{{- 4}}{{5}},\frac{{1}}{{5}},\frac{{27}}{{5}}} \right)\)

CONCEPT:

If A (x1, y1, z1) and C (x2, y2, z2) are two points in a space and let B be a point on the line segment joining A and B such that

  • It divides AC internally in the ratio m:n. Then, the coordinates of B is given by \(\left( {\frac{{m{x_2} + n{x_1}}}{{m + n}},\frac{{m{y_2} + n{y_1}}}{{m + n}},\frac{{m{z_2} + n{z_1}}}{{m + n}}} \right)\)
  • It divides AC externally in the ratio m:n (m ≠ n). Then, the coordinates of B is given by \(\left( {\frac{{m{x_2} - n{x_1}}}{{m - n}},\frac{{m{y_2} - n{y_1}}}{{m - n}},\frac{{m{z_2} - n{z_1}}}{{m - n}}} \right)\)

CALCULATION:

Let us suppose point B divides the line segment joining the points A(- 2, 3, 5) and C(1, - 4, 6) in the ratio 2:3 internally.

Here, x1 = - 2, y1 = 3, z1 = 5, x2 = 1, y2 = - 4, z2 = 6, m = 2 and n = 3

So, the coordinates of B is given by: \(\left( {\frac{{2 - 6}}{{5}},\frac{{-8 + 9}}{{5}},\frac{{12 +15}}{{5}}} \right)\)

⇒ \(B = \left( {\frac{{- 4}}{{5}},\frac{{1}}{{5}},\frac{{27}}{{5}}} \right)\)

Hence, correct option is 4.

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