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Byju's Answer

Standard XI

Economics

Market Supply Curve

Let P x 1, y ...

Question

Let $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ be two points and let R(x, y, z) be a point on PQ dividing it in the ratio m:n. Prove that
$x = \frac{m x_{2} + n x_{1}}{m + n}, y = \frac{m y_{2} + n y_{1}}{m + n}$ and $z = \frac{m z_{2} + n z_{1}}{m + n}$ .

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Solution

From P, Q and R, draw perpendicular PL, QM and RN on the xy-plane. Also, draw $P S ⊥ Q M$ , meeting QM and RN at S and T respectively. From similar triangles PRT and PQS, we have

$\frac{R T}{Q S} = \frac{P R}{P Q}$

$\Rightarrow \frac{R N - T N}{Q M - S M} = \frac{m}{m + n}$

$\Rightarrow \frac{R N - P L}{Q M - P L} = \frac{m}{m + n}$

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{m}{m + n}$

$\Rightarrow z = \frac{m z_{2} + n z_{1}}{m + n}$

Similarly, $x = \frac{m x_{2} + n x_{1}}{m + n}$ and $y = \frac{m y_{2} + n y_{1}}{m + n}$

Hence, $x = \frac{m x_{2} + n x_{1}}{m + n}, y = \frac{m y_{2} + n y_{1}}{m + n}, \frac{m z_{2} + n z_{1}}{m + n}$ .

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Similar questions

The coordinates of the point R which divides the line segment joining two points P (x₁, y₁, z₁) and Q (x₂, y₂, z₂) externally in the ratio m : n are given by

Which of the follwing statements are correct ?
1. The coordinates of the point R which divides the line

segment joining two points $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

externally in the ratio m:n are $(\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n}, \frac{m z_{2} + n z_{1}}{m + n})$ .

2. If R divides PQ internally in the ratio m:n, then its coordinates

are obtained by replacing n by -n in the statement 1.

Q. If P(x,y) divides A

(x_{1}, y_{1})

and B

(x_{2}, y_{2})

in the ratio m : n, then, P(x,y) =

(\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})

Q. Distance of the point

(x_{1}, y_{1}, z_{1})

from the line

\frac{x - x_{2}}{l} = \frac{y - y_{2}}{m} = \frac{z - z_{2}}{n}

, where

l, m

and

n

are the direction cosines of line is

Q. If

A (x_{1}, y_{1}, z_{1})

and

B (x_{2}, y_{2}, z_{2})

are two points such that the direction cosines of

¯ A B

are

l

m

n

then

l = \frac{x_{2} - x_{1}}{∣ ∣ ¯ A B ∣ ∣}

m = \frac{y_{2} - y_{1}}{∣ ∣ ¯ A B ∣ ∣}

n = \frac{z_{2} - z_{1}}{∣ ∣ ¯ A B ∣ ∣}

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Let P(x1, y1, z1) and Q(x2, y2, z2) be two points and let R(x, y, z) be a point on PQ dividing it in the ratio m:n. Prove that x=mx2+nx1m+n, y=my2+ny1m+n and z=mz2+nz1m+n.

Let $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$ be two points and let R(x, y, z) be a point on PQ dividing it in the ratio m:n. Prove that
$x = \frac{m x_{2} + n x_{1}}{m + n}, y = \frac{m y_{2} + n y_{1}}{m + n}$ and $z = \frac{m z_{2} + n z_{1}}{m + n}$ .