Math Problem Statement
In what ratio does the line x – y – 2 = 0 divide the line segment joining (3, –1) and (8, 9)?
Solution
We are tasked with finding the ratio in which the line divides the line segment joining the points and .
Step 1: Equation of the Line
The line equation is , which can be rewritten as:
Step 2: Parametric Coordinates of the Point of Division
Let the line divide the segment in the ratio at a point . The coordinates of dividing and internally in the ratio are given by:
Thus, the coordinates of are .
Step 3: Substituting into the Line Equation
Since lies on the line , we substitute the coordinates of into this equation:
Simplify the left-hand side:
Step 4: Solving for
Now, cross-multiply to eliminate the denominator:
Expand the right-hand side:
Rearrange the terms:
Simplifying:
Final Answer:
The line divides the line segment joining the points and in the ratio 2:3.
Would you like more details or have any questions?
Here are some related questions:
- How do we solve the section formula for external division?
- What is the general method to find the ratio in which a line divides a segment?
- How can we use vectors to solve such ratio problems?
- What happens if the points lie on a vertical or horizontal line?
- Can we find the exact coordinates of the division point in this case?
Tip: For division problems, always ensure whether it is internal or external division—this affects the calculation!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Section Formula
Linear Equations
Formulas
Section formula: Coordinates of point P dividing a line segment internally in ratio m:n are P(x, y) = ((mx2 + nx1) / (m+n), (my2 + ny1) / (m+n))
Equation of line in standard form: Ax + By + C = 0
Theorems
Section Formula Theorem
Suitable Grade Level
Grades 10-12
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