Finding the Ratio for Line Segment Division
This question asks us to determine the ratio in which a specific point, denoted as P(-3, p), divides the line segment formed by joining two other given points, A(-5, -4) and B(-2, 3).
Understanding the Section Formula
To solve this, we utilize the section formula from coordinate geometry. The section formula helps us find the coordinates of a point that divides a line segment internally in a given ratio.
If a point $P(x, y)$ divides the line segment joining points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, the coordinates of P are given by:
In this problem, we are given:
- Point $A(x_1, y_1) = (-5, -4)$
- Point $B(x_2, y_2) = (-2, 3)$
- The dividing point $P(x, y) = (-3, p)$
- The ratio is $m:n$ (which we need to find).
Applying the Formula to Find the Ratio
We can use either the x-coordinate or the y-coordinate of the point P to find the ratio $m:n$. Since the x-coordinate of P is given as -3, let's use the formula for the x-coordinate:
Substitute the known values:
Now, we solve this equation for the ratio $m:n$. Multiply both sides by $(m+n)$:
Rearrange the terms to group m and n together:
To find the ratio $m:n$, we can divide both sides by $-n$ (assuming $n \neq 0$):
This can be written as a ratio:
Conclusion
Therefore, the point (-3, p) divides the line segment joining the points (-5, -4) and (-2, 3) in the ratio 2 : 1. Notice that we didn't need the value of 'p' to determine this ratio.
