Finding 'P' using coordinate geometry

Question

A point 'P' divides the line segment joining the points $A=(3,-5)$ and $B=(-4,8)$ such that $\frac{AP}{PB}=\frac{k}{1}$. If $P$ lies on the line $x+y=0$, then find the value of $k$

Answer 1

Step 1: Find the equation of the straight line $l$ passing through points $A$ and $B$.

Step 2: Find the point $P$ which is the intersection of the straight line $l$ and the straight line $y=-x$.

Step 3: Find the lengths $AP$ and $BP$ and equate their ratios to $k/1$.

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Answer 2

$$P\left(\frac{1\cdot3+k\cdot(-4)}{k+1},\frac{1\cdot(-5)+k\cdot8}{k+1}\right)$$ or $$P\left(\frac{3-4k}{k+1},\frac{8k-5}{k+1}\right).$$ Thus, $$\frac{3-4k}{k+1}+\frac{8k-5}{k+1}=0,$$ which gives $k=\frac{1}{2}$.