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Chapter 14: Problem 1278

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in A.P. then \(\mathrm{ax}+\mathrm{by}+\mathrm{c}=0\) represents (a) a single line (b) a family of concurrent lines (c) a family of parallel lines (d) a family of circle

Short Answer

Expert verified
The given equation represents a family of parallel lines when a, b, and c are in arithmetic progression. The reason is that for any arithmetic progression, the slope of the line will be the same, given by the equation \(-\frac{2\mathrm{a}}{\mathrm{a}+\mathrm{c}}\). Hence, the correct option is (c).

Step by step solution

01

Equation given:

\(\mathrm{ax}+\mathrm{by}+\mathrm{c}=0\) Since a, b, and c are in A.P., this means:
02

Arithmetic Progression condition:

\(\mathrm{b} = \frac{\mathrm{a}+\mathrm{c}}{2}\) Now, we will express the given equation in the slope-intercept form, i.e., y = mx + b, where m is the slope and b is the y-intercept. To accomplish this, we can rewrite the given equation as:
03

Finding the equation in slope-intercept form:

\(\mathrm{y} = -\frac{\mathrm{a}}{\mathrm{b}}\mathrm{x} -\frac{\mathrm{c}}{\mathrm{b}}\) Using the A.P. condition, we can substitute b in the slope-intercept form equation:
04

Substituting b with A.P. condition:

\(\mathrm{y} = -\frac{\mathrm{a}}{\frac{\mathrm{a}+\mathrm{c}}{2}}\mathrm{x} -\frac{2\mathrm{c}}{\mathrm{a}+\mathrm{c}}\) Now, let's analyze the slope and y-intercept: The slope of the line is given by \(-\frac{\mathrm{a}}{\frac{\mathrm{a}+\mathrm{c}}{2}} = -\frac{2\mathrm{a}}{\mathrm{a}+\mathrm{c}}\). From this equation, we can see that for any arithmetic progression, the slope will be the same as the numerator will always be -2a, and the denominator will always be a+c. This indicates that the lines are parallel, and thus our answer is option (c).

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