Answer :
To find the distance between the parallel lines [tex]\( l_1: 2y + x = 3 \)[/tex] and [tex]\( l_2: 2y + x = 1 \)[/tex], you can follow these steps:
1. Recognize the Form of the Lines:
Both lines can be written in the form [tex]\( Ax + By + C = 0 \)[/tex]. So, for:
- Line [tex]\( l_1 \)[/tex]: [tex]\( 2y + x - 3 = 0 \)[/tex], which gives [tex]\( A = 1\)[/tex], [tex]\( B = 2\)[/tex], and [tex]\( C_1 = -3\)[/tex].
- Line [tex]\( l_2 \)[/tex]: [tex]\( 2y + x - 1 = 0 \)[/tex], which gives [tex]\( A = 1\)[/tex], [tex]\( B = 2\)[/tex], and [tex]\( C_2 = -1\)[/tex].
2. Use the Distance Formula for Parallel Lines:
The general formula to calculate the distance [tex]\( d \)[/tex] between two parallel lines of the form [tex]\( Ax + By + C_1 = 0 \)[/tex] and [tex]\( Ax + By + C_2 = 0 \)[/tex] is:
[tex]\[
d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}
\][/tex]
3. Calculate the Difference in Constants:
Find the difference between the constants [tex]\( C_2 \)[/tex] and [tex]\( C_1 \)[/tex]:
[tex]\[
|C_2 - C_1| = |-1 - (-3)| = |2|
\][/tex]
4. Calculate the Denominator:
Compute [tex]\(\sqrt{A^2 + B^2}\)[/tex]:
[tex]\[
\sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\][/tex]
5. Compute the Distance:
Using these values, the distance between the lines is:
[tex]\[
d = \frac{|2|}{\sqrt{5}} = \frac{2}{\sqrt{5}}
\][/tex]
The exact numerical value of the distance [tex]\( \frac{2}{\sqrt{5}} \)[/tex] is approximately [tex]\( 0.894 \)[/tex]. Therefore, the distance between the two parallel lines [tex]\( l_1: 2y + x = 3 \)[/tex] and [tex]\( l_2: 2y + x = 1 \)[/tex] is about 0.894 units.
1. Recognize the Form of the Lines:
Both lines can be written in the form [tex]\( Ax + By + C = 0 \)[/tex]. So, for:
- Line [tex]\( l_1 \)[/tex]: [tex]\( 2y + x - 3 = 0 \)[/tex], which gives [tex]\( A = 1\)[/tex], [tex]\( B = 2\)[/tex], and [tex]\( C_1 = -3\)[/tex].
- Line [tex]\( l_2 \)[/tex]: [tex]\( 2y + x - 1 = 0 \)[/tex], which gives [tex]\( A = 1\)[/tex], [tex]\( B = 2\)[/tex], and [tex]\( C_2 = -1\)[/tex].
2. Use the Distance Formula for Parallel Lines:
The general formula to calculate the distance [tex]\( d \)[/tex] between two parallel lines of the form [tex]\( Ax + By + C_1 = 0 \)[/tex] and [tex]\( Ax + By + C_2 = 0 \)[/tex] is:
[tex]\[
d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}}
\][/tex]
3. Calculate the Difference in Constants:
Find the difference between the constants [tex]\( C_2 \)[/tex] and [tex]\( C_1 \)[/tex]:
[tex]\[
|C_2 - C_1| = |-1 - (-3)| = |2|
\][/tex]
4. Calculate the Denominator:
Compute [tex]\(\sqrt{A^2 + B^2}\)[/tex]:
[tex]\[
\sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}
\][/tex]
5. Compute the Distance:
Using these values, the distance between the lines is:
[tex]\[
d = \frac{|2|}{\sqrt{5}} = \frac{2}{\sqrt{5}}
\][/tex]
The exact numerical value of the distance [tex]\( \frac{2}{\sqrt{5}} \)[/tex] is approximately [tex]\( 0.894 \)[/tex]. Therefore, the distance between the two parallel lines [tex]\( l_1: 2y + x = 3 \)[/tex] and [tex]\( l_2: 2y + x = 1 \)[/tex] is about 0.894 units.
