Question

Find the angle between the line joining the points (2,0), (0,3) and the line \[x+y=1\].

Answer 1

Hint: To solve this type of problem first we have to find the slope of the lines that is formed by the points and the given line. After finding the slope we have to submit in the formula to get the angle formed between the line joining the points and line.

Complete step-by-step answer:
 Let A(2,0) and B(0,3) be the points. Slope of line joining AB is
Formula for slope if two points are given \[{{m}_{1}}=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}\]
\[{{m}_{1}}=\dfrac{3-0}{0-2}\]
\[{{m}_{1}}=\dfrac{-3}{2}\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
Slope of the line \[x+y=1\] is -1.
\[{{m}_{2}}=-1\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
Let \[\theta \] be the angle between the line joining the points (2,0), (3,0) and the line \[x+y=1\].
\[\tan \theta =\left| \dfrac{{{m}_{1}}-{{m}_{2}}}{1+{{m}_{1}}{{m}_{2}}} \right|\]
Substituting the values from (1) and (2), we get
\[\tan \theta \]\[=\left| \dfrac{\dfrac{-3}{2}+1}{1+\dfrac{3}{2}} \right|\] . . . . . . . . . (3)
\[=\left| \dfrac{\dfrac{-3}{2}+1}{1+\dfrac{3}{2}} \right|\]
\[=\dfrac{1}{5}\]
\[\tan \theta =\dfrac{1}{5}\]
\[\theta ={{\tan }^{-1}}\left( \dfrac{1}{5} \right)\]
Hence the angle between the line joining the points (2,0), (3,0) and the line \[x+y=1\] is \[\theta ={{\tan }^{-1}}\left( \dfrac{1}{5} \right)\]

Note: Modulus is placed in (3) because we are calculating the acute angle formed by the lines. Actually we should have got the value negative. By applying modulus we can find the acute angle formed by the lines.