Question

Parthi and Alisha found a treasure that is exactly on the straight line joining their locations. Parthi's location is at point $(-6, -5)$ and Alisha's location is at point $(10, 11)$. The distance from the treasure to Parthi's location is three times that of the distance to Alisha's location. Find the coordinates of the location of the treasure.

Hint:

Double check the starting and ending points when applying the section formula.
Since the ratio is $PT : TA = 3 : 1$, the multiplier 3 is associated with Alisha's coordinates $A(10, 11)$ and the multiplier 1 is associated with Parthi's coordinates $P(-6, -5)$.

Answer 1

Step 1: Think in terms of a fraction of the journey instead of the section formula ratio.
We are told $PT = 3 \times TA$. Since $T$ lies between $P$ and $A$ on the straight line joining them, the whole distance $PA$ splits as $PT + TA$. Writing $TA = t$, we get $PT = 3t$, so
\[ PA = PT + TA = 3t + t = 4t \] This means $T$ is located $\frac{PT}{PA} = \frac{3t}{4t} = \frac{3}{4}$ of the way from $P$ towards $A$.

Step 2: Express the treasure's position using this fraction directly.
If we travel a fraction $k$ of the way from $P(x_1, y_1)$ to $A(x_2, y_2)$, the coordinates reached are
\[ x = x_1 + k(x_2 - x_1), \qquad y = y_1 + k(y_2 - y_1) \] Here $k = \frac{3}{4}$, $P(x_1, y_1) = (-6, -5)$ and $A(x_2, y_2) = (10, 11)$.

Step 3: Compute the $x$-coordinate.
\[ x = -6 + \frac{3}{4}\big(10 - (-6)\big) = -6 + \frac{3}{4}(16) = -6 + 12 = 6 \]
Step 4: Compute the $y$-coordinate.
\[ y = -5 + \frac{3}{4}\big(11 - (-5)\big) = -5 + \frac{3}{4}(16) = -5 + 12 = 7 \]
Final Answer:
The coordinates of the location of the treasure are $(6, 7)$. \[ \boxed{(6, 7)} \]