Question

What are the coordinates of the point on the directed line segment from (-4, 2) to

(-1,-4) that partitions the segment into a ratio of 1 to 5?

Answer 1

Answer: M(-3.5,1)

Explanation:

`If \ two\ points \ of \ the\ plane \ are\ known:\ A(x_A,y_B) \ and\ B(x_B,y_B)\ \ \ \ \ \ \`

`then\ the \ coordinates \ of \ the \ point\ M(x_M,y_M)`

`\displaystyle \\which \ divides\ the\ segment \ in\ the\ ratio\ \lambda=(AM)/(BM)`

`are \ expressed \ by \ the \ formulae:`

`\displaystyle \\\boxed { x_M=(x_A+\lambda x_B)/(1+\lambda)\ \ \ \ \ \ \ \ \ \ y_M=(y_A+\lambda y_B)/(1+\lambda) }`

A(-4,2) B(-1,-4) λ=1/5 M(x,y)=?

`\displaystyle \\x_A=-4\ \ \ \ x_B=-1\ \ \ \ y_A=2\ \ \ \ y_B=-4\\\\\displaystyle \\x_M=(-4+(1)/(5)*(-1) )/(1+(1)/(5) ) \\\\x_M=(-4-(1)/(5) )/(1(1)/(5) ) \\\\x_M=(-4(1)/(5) )/((6)/(5) ) \\\\x_M=-((21)/(5) )/((6)/(5) ) \\\\x_M=-(21)/(6) \\\\x_M=-3.5\\\\\displaystyle \\y_M=(2+(1)/(5)*(-4) )/(1+(1)/(5) ) \\\\y_M=(2-(4)/(5) )/(1(1)/(5) ) \\\\y_M=(1(1)/(5) )/(1(1)/(5) ) \\\\y_M=1\\\\Hence M(-3.5;1)`

Answer 2

Therefore, the coordinates of the required point are (-13/6, -3).

Explanation:

To find the coordinates of the point on the directed line segment from (-4, 2) to (-1,-4) that partitions the segment into a ratio of 1 to 5, we can use the concept of section formula.

Let's assume that the required point divides the given line segment in the ratio of 1:5. Therefore, let's consider that this point divides the line segment into two parts, one part is x times longer than the other part.

According to section formula, if a line segment joining two points (x1, y1) and (x2, y2) is divided by a point (x, y) in the ratio m:n, then the coordinates of the point (x, y) are given by:

x = (nx2 + mx1)/(m+n)

y = (ny2 + my1)/(m+n)

Using this formula and substituting the given values, we get:

x = (5*(-1) + 1*(-4))/(1+5) = -13/6

y = (5*(-4) + 1*2)/(1+5) = -18/6 = -3