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: Understanding the Question:
The locations of Parthi and Alisha are given by coordinates $P(-6, -5)$ and $A(10, 11)$, respectively.
A treasure at point $T(x, y)$ lies on the straight line segment joining $P$ and $A$.
The distance from the treasure to Parthi ($PT$) is three times the distance from the treasure to Alisha ($TA$).
We need to find the coordinates $(x, y)$ of the treasure.

Step 2: Key Formula or Approach:
Since $PT = 3 \cdot TA$, the ratio of the distances is:
\[ \frac{PT}{TA} = \frac{3}{1} \]
This means the point $T(x, y)$ divides the line segment $PA$ internally in the ratio $m_1 : m_2 = 3 : 1$.
By the Section Formula, the coordinates of a point dividing the line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ in the ratio $m_1 : m_2$ internally are:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]

Step 3: Detailed Explanation:

• 1. Identify the given coordinates and ratio parameters:
- $P(x_1, y_1) = (-6, -5)$
- $A(x_2, y_2) = (10, 11)$
- Ratio $m_1 : m_2 = 3 : 1$

• 2. Calculate the $x$-coordinate of the treasure $T$:
\[ x = \frac{3(10) + 1(-6)}{3 + 1} \]
\[ x = \frac{30 - 6}{4} \]
\[ x = \frac{24}{4} = 6 \]

• 3. Calculate the $y$-coordinate of the treasure $T$:
\[ y = \frac{3(11) + 1(-5)}{3 + 1} \]
\[ y = \frac{33 - 5}{4} \]
\[ y = \frac{28}{4} = 7 \]


Step 4: Final Answer:
The coordinates of the location of the treasure are $(6, 7)$.