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 xy2=0x - y - 2 = 0 divides the line segment joining the points A(3,1)A(3, -1) and B(8,9)B(8, 9).

Step 1: Equation of the Line

The line equation is xy2=0x - y - 2 = 0, which can be rewritten as:

xy=2x - y = 2

Step 2: Parametric Coordinates of the Point of Division

Let the line divide the segment in the ratio m:nm:n at a point P(x1,y1)P(x_1, y_1). The coordinates of PP dividing A(3,1)A(3, -1) and B(8,9)B(8, 9) internally in the ratio m:nm:n are given by:

x1=mxB+nxAm+n=m8+n3m+nx_1 = \frac{m \cdot x_B + n \cdot x_A}{m + n} = \frac{m \cdot 8 + n \cdot 3}{m + n}

y1=myB+nyAm+n=m9+n(1)m+ny_1 = \frac{m \cdot y_B + n \cdot y_A}{m + n} = \frac{m \cdot 9 + n \cdot (-1)}{m + n}

Thus, the coordinates of PP are (8m+3nm+n,9mnm+n)\left( \frac{8m + 3n}{m + n}, \frac{9m - n}{m + n} \right).

Step 3: Substituting into the Line Equation

Since PP lies on the line xy=2x - y = 2, we substitute the coordinates of PP into this equation:

8m+3nm+n9mnm+n=2\frac{8m + 3n}{m + n} - \frac{9m - n}{m + n} = 2

Simplify the left-hand side:

(8m+3n)(9mn)m+n=2\frac{(8m + 3n) - (9m - n)}{m + n} = 2

8m+3n9m+nm+n=2\frac{8m + 3n - 9m + n}{m + n} = 2

m+4nm+n=2\frac{-m + 4n}{m + n} = 2

Step 4: Solving for m:nm:n

Now, cross-multiply to eliminate the denominator:

m+4n=2(m+n)-m + 4n = 2(m + n)

Expand the right-hand side:

m+4n=2m+2n-m + 4n = 2m + 2n

Rearrange the terms:

m2m=2n4n-m - 2m = 2n - 4n

3m=2n-3m = -2n

Simplifying:

m:n=2:3m:n = 2:3

Final Answer:

The line xy2=0x - y - 2 = 0 divides the line segment joining the points (3,1)(3, -1) and (8,9)(8, 9) in the ratio 2:3.

Would you like more details or have any questions?

Here are some related questions:

  1. How do we solve the section formula for external division?
  2. What is the general method to find the ratio in which a line divides a segment?
  3. How can we use vectors to solve such ratio problems?
  4. What happens if the points lie on a vertical or horizontal line?
  5. 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