Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio $2 : 3$ , and Q divides AB in the ratio 3:4. If PQ=2, then find the length of the line segment AB.

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Solution

$P$ divides of the line $A B = 2 : 3$
$Q$ divides of line $A B = 3 : 4$
The distance between $P$ and $Q = 2 c m$
we have to find the length of $A B = ?$
$A B = A P + P B = 2 x + 3 x = 5 x$
Or,
$A B = A Q + Q B = 3 y + 4 y = 7 y$

Now, $A Q = A P + P Q$
$3 y = 2 x + 2$
$3 y - 2 x = 2 \dots \dots (i)$
$B P = B Q + 2$
$3 x = 4 y + 2 \dots \dots (i i)$
$3 x - 4 y = 2 \dots \dots (i i)$
On multiplying (i) by 3 and eq. (ii), by 2, after that adding both, we get
$9 y - 6 x + (6 x - 8 y) = 6 + 4$
$\Rightarrow y = 10$
So, $A B = 7 y = 7 \times 10 = 70 c m$

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Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio 2:3, and Q divides AB in the ratio 3:4. If PQ=2, then find the length of the line segment AB.

Points P and Q are both in the line segment AB and on the same side of its midpoint. P divides AB in the ratio $2 : 3$ , and Q divides AB in the ratio 3:4. If PQ=2, then find the length of the line segment AB.