Question

city planning the united states capitol is located at (2, -4) on a coordinate grid. the white house is located at (-10, 16) on the same coordinate grid. find the two points on the straight line between the united states capitol and the white house such that the ratio is 1:3.
a) (7, 11) and (1, 1)
b) (-7, 11) and (-1, 1)
c) (-7, -11) and (-1, -1)
d) (7, -11) and (1, -1)

Explanation:

Step1: Recall Section Formula

The section formula for a point \((x,y)\) dividing the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2 + nx_1}{m + n}\), \(y=\frac{my_2 + ny_1}{m + n}\). Here, Capitol is \((x_1,y_1)=(2,-4)\), White House is \((x_2,y_2)=(-10,16)\), ratio \(m:n = 1:3\).

Step2: Calculate x-coordinate

Substitute into \(x\)-formula: \(x=\frac{1\times(-10)+3\times2}{1 + 3}=\frac{-10 + 6}{4}=\frac{-4}{4}=-1\)

Step3: Calculate y-coordinate

Substitute into \(y\)-formula: \(y=\frac{1\times16+3\times(-4)}{1 + 3}=\frac{16-12}{4}=\frac{4}{4}=1\)

Wait, but maybe the ratio is reversed? Wait, the problem says "two points on the straight line between the United States Capitol and the White House such that the ratio is 1:3". So maybe the points are dividing the segment into parts with ratio 1:3, so two points: one closer to Capitol, one closer to White House? Wait, no, the options have two points. Wait, maybe we need to find two points that divide the segment in ratio 1:3 and 3:1? Wait, let's re - evaluate.

Wait, maybe the first point is dividing Capitol to White House in 1:3, and the second is dividing White House to Capitol in 1:3 (i.e., 3:1 from Capitol to White House).

First, for ratio \(m:n = 1:3\) (Capitol to White House):

\(x_1=\frac{1\times(-10)+3\times2}{4}=\frac{-10 + 6}{4}=-1\), \(y_1=\frac{1\times16+3\times(-4)}{4}=\frac{16 - 12}{4}=1\)

Second, for ratio \(m:n = 3:1\) (Capitol to White House):

\(x_2=\frac{3\times(-10)+1\times2}{4}=\frac{-30 + 2}{4}=\frac{-28}{4}=-7\), \(y_2=\frac{3\times16+1\times(-4)}{4}=\frac{48-4}{4}=\frac{44}{4}=11\)

So the two points are \((-7,11)\) and \((-1,1)\), which is option B.

Answer:

B) \((-7,11)\) and \((-1,1)\)