Finding Values of a and b for a Straight Line Through Given Points

To find the values of (a) and (b) for the line given by the equation ( frac{x}{a} frac{y}{b} 1 ), follow these steps:

Solution Method

First, let's rewrite the given line equation in slope-intercept form (y mx c). We start by rearranging the terms:

[y -frac{b}{a}x b]

This shows that the slope (m) of the line is (-frac{b}{a}) and the y-intercept is (b).

Step 1: Calculate the Slope Between Given Points

Given points are (8, -9) and (12, -15). The slope (m) between two points ((x_1, y_1)) and ((x_2, y_2)) is:

[m frac{y_2 - y_1}{x_2 - x_1}]

Substituting the points (8, -9) and (12, -15):

[m frac{-15 - (-9)}{12 - 8} frac{-15 9}{4} frac{-6}{4} -frac{3}{2}]

Step 2: Relate the Slope to (a) and (b)

From the line equation (-frac{b}{a} -frac{3}{2}), we get:

[frac{b}{a} frac{3}{2} implies b frac{3}{2}a]

Step 3: Find the Y-Intercept

We can find (b) by substituting one of the points into the line equation. Let's use the point (8, -9):

[frac{8}{a} frac{-9}{b} 1]

Substituting (b frac{3}{2}a):

[frac{8}{a} - frac{9}{frac{3}{2}a} 1]

This simplifies to:

[frac{8}{a} - frac{9 cdot 2}{3a} 1 frac{8}{a} - frac{18}{3a} 1 frac{8}{a} - frac{6}{a} 1 frac{2}{a} 1 a 2]

Step 4: Calculate (b)

Using (a 2) in the equation (b frac{3}{2}a):

[b frac{3}{2} cdot 2 3]

Conclusion

The values of (a) and (b) are:

[boxed{2} text{ and } boxed{3}]

Additional Methods for Verification:

Method 1: Using Algebraic Equations

The equation of the straight line passing through (8, -9) and (12, -15) is ( frac{x}{a} frac{y}{b} 1 ). Given points, substitute (8, -9) and (12, -15):

[frac{8}{a} - frac{9}{b} 1]

Or when multiplied by (ab):

[8b - 9a ab]

Similarly for (12, -15):

[12b - 15a ab]

Subtracting the first equation from the second:

[12b - 15a - (8b - 9a) ab - ab]

This simplifies to:

[12b - 15a - 8b 9a 0 4b - 6a 0 4b 6a implies b frac{3}{2}a]

Substituting (b frac{3}{2}a) in the first equation:

[frac{8}{a} - frac{9}{frac{3}{2}a} 1 frac{8}{a} - frac{9 cdot 2}{3a} 1 frac{8}{a} - frac{18}{3a} 1 frac{8}{a} - frac{6}{a} 1 frac{2}{a} 1 a 2]

In conclusion, (a 2) and (b 3).

Note: If (a 0), the values (a 0) and (b 0) are also possible, but the line equation would not be valid in such cases.