Question

If the coordinates of the points A and B are (-2,-2) and (2,-4) respectively, find the coordinates of P such that $AP = \dfrac{3}{7}AB$, where P lies on the line segment AB.

Answer 1

Hint: To get the coordinates of P, first we will solve the given equation to get the ratio by which the point P divides the line segment AB. Then considering that ratio with m:n, we will get the value of m and n. Putting m and n and the coordinates of A and B in the section rule, we will obtain the coordinates of P.

Complete step-by-step answer:
According to the question, the coordinates of point A is (-2,-2)
The coordinates of point B is (2,-4)
P lies on the line segment AB such that
$AP = \dfrac{3}{7}AB$
Representing the above data in a diagram,

By doing cross multiplication we get,
$7AP = 3AB$
As P lies on the line segment AB then we can write AB as the sum of AP and PB, i.e.
$7AP = 3(AP + PB)$
Expanding the above equation and taking all the AP term to the left hand side we get,
$7AP - 3AP = 3PB$
$ \Rightarrow 4AP = 3PB$
Taking PB to the denominator of AP and 4 to the denominator of 3 we will get the ratio of AP and PB,
$\dfrac{{AP}}{{PB}} = \dfrac{3}{4}$……….(1)
From the above the equation we get that point P divides line segment AB in the ratio 3:4
Let the coordinates of P be (a, b) and it divides the line segment in m:n ratio. i.e. m=3 and n=4
According to the section rule, if a point (a, b) divides the line joining the point (x, y) and the point (x’, y’) in the ratio m:n then the coordinates of the point is given as,
$(a,b) = \left( {\dfrac{{mx' + nx}}{{m + n}},\dfrac{{my' + ny}}{{m + n}}} \right)$
Hence putting P (a, b), A (-2,-2), B (2,-4) and the value of m and n in the above formula we get,
The coordinates of P is $P(a,b) = \left( {\dfrac{{3 \times 2 + 4 \times ( - 2)}}{{3 + 4}},\dfrac{{3 \times ( - 4) + 4 \times ( - 2)}}{{3 + 4}}} \right)$
Simplifying the above equation we get,
$P(a,b) = \left( {\dfrac{{6 + ( - 8)}}{7},\dfrac{{( - 12) + ( - 8)}}{7}} \right)$
$ \Rightarrow P(a,b) = \left( {\dfrac{{( - 2)}}{7},\dfrac{{( - 20)}}{7}} \right)$
Hence the coordinates of P are $\left( {\dfrac{{( - 2)}}{7},\dfrac{{( - 20)}}{7}} \right)$

Note: You might confuse why P divides AB internally. It is because AB is the line segment and P lies on it. If AB was a line then we would have to obtain whether P divides it externally or internally.
A line segment has two endpoints but the line has no end points.
The section rule states that if a point (a, b) divides the line joining the point (x, y) and the point (x’, y’) in the ratio m:n then the coordinates of the point is given as,
$(a,b) = \left( {\dfrac{{mx' + nx}}{{m + n}},\dfrac{{my' + ny}}{{m + n}}} \right)$
You should remember all the rules and formulae of coordinate geometry.