Question

If the image of the point $ \left( -1,2 \right) $ in the line $ 4x+3y-p=0 $ is the point $ \left( 3,5 \right) $ then find the value of $ p $ .

Answer 1

Hint: Here we have been given a point whose image in a line given is a point and we have to find the value of an unknown variable in the line equation. As we know that since the line is acting as a mirror for the two points given so the mid-point obtained from both the points will lie in the line and that point will satisfy the equation of the line using this concept we will find our answer.

Complete step-by-step answer:
The point whose image is given is as follows:
  $ \left( -1,2 \right) $
Let us denote it as follows:
 $ A=\left( -1,2 \right) $
The image of the above point is given as:
 $ \left( 3,5 \right) $
Let us denote it as follows:
 $ B=\left( 3,5 \right) $
The two points will be plotted in the graph as follows:

The line with respect to which point $ A $ image is given is as follows:
 $ 4x+3y-p=0 $ …… $ \left( 1 \right) $
So our first step will be to find the midpoint of point $ A $ and $ B $
The midpoint formula is given as follows:
\[\left( \dfrac{{{x}_{1}}+{{x}_{2}}}{2},\dfrac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]….. $ \left( 2 \right) $
As $ A=\left( -1,2 \right)=\left( {{x}_{1}},{{y}_{1}} \right) $ and $ B=\left( 3,5 \right)=\left( {{x}_{2}},{{y}_{2}} \right) $
Substitute the above value in equation (2) and simplify as follows:
 $ \Rightarrow \left( \dfrac{-1+3}{2},\dfrac{2+5}{2} \right) $
 $ \Rightarrow \left( 1,\dfrac{7}{2} \right) $
Now as we know that the mi-point of the two points will lie in the line given we will substitute the above point in equation (1) and simplify as follows:
 $ \Rightarrow 4\times 1+3\times \dfrac{7}{2}-p=0 $
 $ \Rightarrow 4+\dfrac{21}{2}-p=0 $
Take the unknown variable on one side and rest value on another as follows:
 $ \Rightarrow p=4+\dfrac{21}{2} $
 $ \Rightarrow p=\dfrac{8+21}{2} $
So we get the value as:
 $ p=\dfrac{29}{2} $
Hence the image of the point $ \left( -1,2 \right) $ in the line $ 4x+3y-p=0 $ is the point $ \left( 3,5 \right) $ when the value of $ p $ is $ \dfrac{29}{2} $ .
So, the correct answer is “ $ \dfrac{29}{2} $ ”.

Note: Midpoint formula is used to find the center point of a straight line of midpoint of two points. The $ x $ coordinate point and $ y $ coordinate point are calculated separately. It is the average of the two numbers of the two points. We have to be careful while taking the value of $ {{x}_{1}},{{y}_{1}},{{x}_{2}},{{y}_{2}} $ . We can cross check our answer by using the help of a graph as follows: