Question

14 line segment ab has endpoint a located at the origin. line segment ab is longest when the coordinates of b are 1) (3,7) 2) (2, - 8) 3) (-6,4) 4) (-5,-5)

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Since $A=(0,0)$, for a point $B=(x,y)$, the distance $d=\sqrt{x^{2}+y^{2}}$.

Step2: Calculate distance for option 1

For $B=(3,7)$, $d_1=\sqrt{3^{2}+7^{2}}=\sqrt{9 + 49}=\sqrt{58}$.

Step3: Calculate distance for option 2

For $B=(2,-8)$, $d_2=\sqrt{2^{2}+(-8)^{2}}=\sqrt{4 + 64}=\sqrt{68}$.

Step4: Calculate distance for option 3

For $B=(-6,4)$, $d_3=\sqrt{(-6)^{2}+4^{2}}=\sqrt{36+16}=\sqrt{52}$.

Step5: Calculate distance for option 4

For $B=(-5,-5)$, $d_4=\sqrt{(-5)^{2}+(-5)^{2}}=\sqrt{25 + 25}=\sqrt{50}$.

Step6: Compare distances

Since $\sqrt{68}>\sqrt{58}>\sqrt{52}>\sqrt{50}$, the longest distance is for the point $(2,-8)$.

Answer:

1. $(2,-8)$