Certainly! Let's solve the problem.
We are given the line equation [tex]\(3x + y - 9 = 0\)[/tex] which divides the line segment joining the points [tex]\( A(1, 3) \)[/tex] and [tex]\( B(2, 7) \)[/tex]. We need to find the ratio in which this line divides the segment [tex]\( AB \)[/tex].
### Step-by-Step Solution:
1. Representing the Points and Line:
- Line: [tex]\(3x + y - 9 = 0\)[/tex]
- Point [tex]\( A \)[/tex]: [tex]\((1, 3)\)[/tex]
- Point [tex]\( B \)[/tex]: [tex]\((2, 7)\)[/tex]
2. Calculate the Distance from Point [tex]\( A \)[/tex] to the Line:
- General formula for the distance from a point [tex]\( (x_1, y_1) \)[/tex] to the line [tex]\( ax + by + c = 0 \)[/tex] is:
[tex]\[
d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}
\][/tex]
- Substituting [tex]\( A(1, 3) \)[/tex] into the formula where [tex]\( a = 3 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -9 \)[/tex]:
[tex]\[
d_1 = \frac{|3(1) + 1(3) - 9|}{\sqrt{3^2 + 1^2}} = \frac{|3 + 3 - 9|}{\sqrt{9 + 1}} = \frac{|6 - 9|}{\sqrt{10}} = \frac{|-3|}{\sqrt{10}} = \frac{3}{\sqrt{10}}
\][/tex]
- Simplifying further gives:
[tex]\[
d_1 \approx 0.9487
\][/tex]
3. Calculate the Distance from Point [tex]\( B \)[/tex] to the Line:
- Substituting [tex]\( B(2, 7) \)[/tex] into the same distance formula:
[tex]\[
d_2 = \frac{|3(2) + 1(7) - 9|}{\sqrt{3^2 + 1^2}} = \frac{|6 + 7 - 9|}{\sqrt{9 + 1}} = \frac{|13 - 9|}{\sqrt{10}} = \frac{4}{\sqrt{10}}
\][/tex]
- Simplifying further gives:
[tex]\[
d_2 \approx 1.2649
\][/tex]
4. Calculate the Ratio in which the Line Divides the Segment [tex]\( AB \)[/tex]:
- The ratio [tex]\( k \)[/tex] in which the line divides the segment [tex]\( AB \)[/tex] can be found by the ratio of these distances:
[tex]\[
k = \frac{d_1}{d_2} = \frac{0.9487}{1.2649} \approx 0.75
\][/tex]
Therefore, the line [tex]\( 3x + y - 9 = 0 \)[/tex] divides the line segment joining the points [tex]\((1, 3)\)[/tex] and [tex]\((2, 7)\)[/tex] in the ratio [tex]\( \boxed{0.75} \)[/tex].
