Question

If P(a,b) divides B(h,k) and C(l,m) such that $a$ = $\frac{pl+qh}{p+q}$. Find the ratio in which P divides BC.

• A. l : h
• B. p : q
• C. q : p
• D. k : m

Answer 1

The correct option is B

p : q

Let us say point P(a,b) divides the line BC in ratio x:y
The coordinate of the point that divides a line joining two points $(x_1,y_2),(x_2,y_2)$ in the ratio m:n is,

$(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

Here substituting the points B(h,k) and C(l,m) we get

$P(a,b)=(\frac{xl+yh}{x+y},\frac{xm+yk}{x+y})\therefore a=\frac{xl+yh}{x+y}\dots \left(1\right)\text{But given that}a=\frac{pl+qh}{p+q}\dots \left(2\right)$

Comparing (1) and (2) we can say that $x=p\text{and}y=q$

Hence P divides BC in the ratio p:q.