Question

A point R with X coordinate 4 liEs On the line segment Joining the pOint P(2,-3,4) ND Q (8,0,10) find the coordinate of the point R .

Answer 1

let the line segment divides in m:1
it is given that the x coordinate of point R is 4
therefore
8m+2/m+1=4
=m=1/2
therefore point R
=(4,-3/(1/2+1),(10(1/2)+4)/(1/2+1))
=(4,-2,6)

Answer 2

The coordinates of point R is (4,-2,6).

• A point R lies on the segment PQ whose x-coordinate is 4.

Let point R be (4,a,b).

• Let point R divide the line segment PR in the ratio k : 1.

The coordinate of the point that divides the line segment joining (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio m:n is

$(\frac{mx_2 + nx_1}{m+n} , \frac{my_2 + ny_1}{m+n} , \frac{mz_2 + nz_1}{m+n})$

Here the ratio is k:1.

Therefore, m=k and n=1

x₁=2  and y₁=-3

x₂=8  and y₂=0

• Putting values in the formula,

R = $(\frac{k(8) + 1(2)}{k+1} , \frac{k(0) + 1(-3)}{k+1} , \frac{k(10) + 1(4)}{k+1})$

(4,a,b) = $(\frac{8k + 2}{k+1} , \frac{-3}{k+1} , \frac{10k+4}{k+1})$

Equating both sides we get,

• for x-coordinate,

4 = $\frac{8k+2}{k+1}$

4k+4 = 8k+2

4k = 2

k = 1/2.

Now we substitute the value of k in y-coordinate and z-coordinate to find a and b

• Now y-coordinate is

b = $\frac{-3}{k+1}$

b(k+1) = -3

b(1/2+1) = -3

3b = -6

b = -2

• Now z-coordinate ,

c = $\frac{10k+4}{k+1}$

c(k+1) = (10k+4)

3c = 18

c = 6

• The coordinates of R is (4,-2,6).