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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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Most popular questions from this chapter

A line intersects \(\mathrm{X}\) -axis and Y-axis at \(\mathrm{A}\) and \(\mathrm{B}\) respectively. If \(\mathrm{AB}=15\) and \(\underline{\mathrm{AB}}\) makes a triangle of area 54 units with coordinate axes, then the equation of \(\underline{A B}\) is \(\ldots .\) (a) \(4 x \pm 3 y=36\) or \(3 x \pm 4 y=36\) (b) \(4 x \pm 3 y=24\) or \(3 x \pm 4 y=24\) (c) \(-4 \mathrm{x} \pm 3 \mathrm{y}=24\) or \(-3 \mathrm{x} \pm 4 \mathrm{y}=24\) (d) \(-4 x \pm 3 y=12\) or \(-3 x \pm 4 y=12\)

If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}\) and \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}\) are in geometric progression and their common ratios are equal, then the points \(\mathrm{A}\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right)\) \(\mathrm{B}\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{a}_{3}, \mathrm{~b}_{3}\right) \ldots \ldots\) (a) lie on the same line (b) lie on a circle (c) lie on an ellipse (d) None of these

Find the slope of the line passing through the point \((1,2)\) and the point of intersection of this line with the line \(\mathrm{x}+\mathrm{y}+3=0\) is at a distance \(3 \sqrt{2}\) units from \((1,2)\). (a) \((1 / \sqrt{3})\) (b) 1 (c) \(\sqrt{3}\) (d) \([(\sqrt{3}-1) / 2]\)

If \((1 / a),(1 / b),(1 / c)\) are in arithmetic sequence, then the line \((\mathrm{x} / \mathrm{a})+(\mathrm{y} / \mathrm{b})+(1 / \mathrm{c})=0\) passes through the fixed point \(\ldots \ldots\) (a) \((-1,-2)\) (b) \((-1,2)\) (c) \([1,-(1 / 2)]\) (d) \((1,-2)\)

The line parallel to the \(X\) -axis and passing through the intersection of the lines \(\mathrm{ax}+2 \mathrm{by}+3 \mathrm{~b}=0\) and \(\mathrm{bx}-2 \mathrm{ay}-3 \mathrm{a}=0\) where \((\mathrm{a}, \mathrm{b}) \neq(0,0)\) is: (A) above the X-axis at a distance of \((2 / 3)\) from it (B) above the X-axis at a distance of \((3 / 2)\) from it (C) below the X-axis at a distance of \((2 / 3)\) from it (D) below the X-axis at a distance of \((3 / 2)\) from it
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