Question

On a coordinate grid PQ , has the endpoints -P(-2, -11) ans Q(13,4) What is the

location of a point R on PQ that is two-fifths of the way from P to Q?

Answers in pic

Answer 1

Answer: A. ( 16/7, -47/7)

Explanation:

Since, when a point divides a line having end points
`(x_1,y_1)` and
`(x_2,y_2)`,

in the ratio of m:n,

Then, by the section formula,

The coordinates of the point are,

`((mx_2+nx_1)/(m+n), (my_2+ny_1)/(m+n))`

Here, a line having end points P(-2, -11) and Q(13,4) is divided by R in the ratio of 2:5,

⇒
`x_1 = -2`,
`y_1 = -11`,
`x_2 = 13`,
`y_2 = 4`,
`m = 2`,
`n = 5`,

Hence, the coordinates of point R

=
`((2* -2+5* -11)/(2+5), (2* 4+5* -11)/(2+5))`

=
`((-4-55)/(7),(8+-55)/(7))`

=
`((16)/(7),(-47)/(7))`

⇒ Option A is correct.

Answer 2

B. (4,-5)

Explanation:

We are given the end-points P = (-2,-11) and Q = (13,4).

The point R is located two-fifths on the way from P to Q.

So, we get that, the ratio sides PQ and QR is 2 : 3

Using the Ratio Formula, we have,

The co-ordinates of the point between
`(x_(1),y_(1))` and (
`(x_(2),y_(2))` having ratio
`m_(1):m_(2)` are given by,

`((m_(1)x_(2)+m_(2)x_(1))/(m_(1)+m_(2)),(m_(1)y_(2)+m_(2)y_(1))/(m_(1)+m_(2)))`

That is we have,

R =
`((2* 13+3* (-2))/(2+3),(2* 4+3* (-11))/(2+3))`

i.e. R =
`((26-6)/(5),(8-33)/(5))`

i.e. R =
`((20)/(5),(-25)/(5))`

i.e. R = (4,-5)

Thus, the co-ordinates of R is (4,-5).

Hence, option B is correct.