Question

Point P divides the line segment joining the points A(2, 1) and B (5, −8) such that $ \dfrac{AP}{AB}=\dfrac{1}{3} $ ​. If P lies on the line 2x – y + k = 0, find the value of k.

Answer 1

Hint: First, find reciprocal of $ \dfrac{AP}{AB}=\dfrac{1}{3} $ . Then rewrite AB as AP + BP. To get $ \dfrac{BP}{AP}=2 $ . Again, take its reciprocal to get $ \dfrac{AP}{BP}=\dfrac{1}{2} $ . Now, use the section formula to get the coordinates of $ P=\left( \dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n} \right) $ . Substitute the resulting coordinates in 2x – y + k = 0. Solve this to find the value of k which is the final answer.

Complete step-by-step answer:
In this question, we are given that a point P divides the line segment joining the points  A (2, 1) and B (5, −8) such that  $ \dfrac{AP}{AB}=\dfrac{1}{3} $ ​.
If P lies on the line 2x – y + k = 0, we need to find the value of k.
We are given that $ \dfrac{AP}{AB}=\dfrac{1}{3} $
We will take its reciprocal. Doing this, we will get the following:
 $ \dfrac{AB}{AP}=3 $
Now, AB can be written as AP + BP. Substituting this in the above expression, we will get the following:
 $ \dfrac{AP+BP}{AP}=3 $
Now, separating the numerator such that we have two terms in the LHS, we will have the following:
 $ 1+\dfrac{BP}{AP}=3 $
 $ \dfrac{BP}{AP}=2 $
Taking the reciprocal of this, we will get the following:
 $ \dfrac{AP}{BP}=\dfrac{1}{2} $
So, the point P divides the line segment AB in the ratio 1 : 2.
Now we will use the section formula to find the coordinates of the point P.
If point P (x, y) lies on the line segment AB and satisfies AP : PB = m :: n, then we say that P divides AB internally in the ratio m : n. The point of division, P has the coordinates:
 $ P=\left( \dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n} \right) $ , where A (a, b) and B (c, d).
So, using the section formula, we will get the following:
 $ P=\left( \dfrac{5\times 1+2\times 2}{2+1},\dfrac{-8\times 1+2\times 1}{2+1} \right) $
 $ P=\left( \dfrac{9}{3},\dfrac{-6}{3} \right)=\left( 3,-2 \right) $
Now, this point P lies on the line 2x – y + k = 0. Substituting $ P=\left( 3,-2 \right) $ in 2x – y + k = 0, we will get the following:
 $ 2\times 3-\left( -2 \right)+k=0 $
 $ 6+2+k=0 $
 $ k=-8 $
This is our final answer.

Note: In this question, it is very important to know about the section formula. The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m : n. If point P (x, y) lies on the line segment AB and satisfies AP : PB = m :: n, then we say that P divides AB internally in the ratio m : n. The point of division, P has the coordinates: $ P=\left( \dfrac{mc+na}{m+n},\dfrac{md+nb}{m+n} \right) $ , where A (a, b) and B (c, d).