Question

chapter ii.4 score: 1/5 answered: 3/5 question 4 the line segment l has endpoints (-13, -12) and (-7, 3). determine the point p in l so that the distance from (-13, -12) to p is 10 times the distance from p to (-7, 3).

Explanation:

Step1: Use section - formula

Let the endpoints of the line - segment be $A(-13,-12)$ and $B(-7,3)$, and let the point $P(x,y)$ divide the line - segment $AB$ in the ratio $m:n = 10:1$.
The section formula for a point $(x,y)$ that divides the line - segment joining $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$ is given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$.

Step2: Calculate the x - coordinate of point $P$

Here, $x_1=-13$, $x_2=-7$, $m = 10$, and $n = 1$.
$x=\frac{10\times(-7)+1\times(-13)}{10 + 1}=\frac{-70-13}{11}=\frac{-83}{11}\approx - 7.55$.

Step3: Calculate the y - coordinate of point $P$

Here, $y_1=-12$, $y_2 = 3$, $m = 10$, and $n = 1$.
$y=\frac{10\times3+1\times(-12)}{10 + 1}=\frac{30-12}{11}=\frac{18}{11}\approx1.64$.

Answer:

$(-\frac{83}{11},\frac{18}{11})$