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Chapter 14: Problem 1329

If \(\left(\mathrm{a}, \mathrm{a}^{2}\right)\) falls inside the angle made by the lines \(\mathrm{y}=(\mathrm{x} / 2), \mathrm{x}>0\) and \(\mathrm{y}=3 \mathrm{x}, \mathrm{x}>0\) the 'a' belongs to (a) \((3, \infty)\) (b) \([(1 / 2), 3]\) (c) \([(-3),-(1 / 2)]\) (d) \([0,(1 / 2)]\)

Short Answer

Expert verified
The 'a' belongs to the interval \((\frac{1}{2}, 3)\), which corresponds to answer choice (b).

Step by step solution

01

Graph and Intersection of Lines

We are given two lines: \(y = \frac{x}{2}\) where \(x > 0\) \(y = 3x\) where \(x > 0\) These lines both pass through the origin (0, 0). To find the intersection of these lines, we'll set the equations equal to each other: \(\frac{x}{2} = 3x\) Solve for x: \(x = \frac{1}{5}\) Now, we find the y-coordinate of the intersection. \(y = 3(\frac{1}{5}) = \frac{3}{5}\) So the point of intersection is (\(\frac{1}{5}, \frac{3}{5}\)).
02

Determine Which Line Is Above the Other

This will help us understand the angle between the lines. To do this, we can evaluate how y-values change as x-values increase for each line. We can see that as \(x\) increases, the slope of the line \(y = 3x\) also increases faster than the line \(y = \frac{x}{2}\). Therefore, the line \(y = 3x\) is above the line \(y = \frac{x}{2}\) when \(x > 0\).
03

Determine the Conditions for the Point to Fall Inside the Angle

For the point \((a, a^2)\) to fall inside the angle, it needs to be above the lower line \(\frac{x}{2}\) and below the upper line \(3x\). This means that the following conditions must be met: \(a^2 > \frac{a}{2}\) \(a^2 < 3a\)
04

Solving the Inequalities

First inequality: \(a^2 > \frac{a}{2}\) Multiplying both sides of the inequality by 2, we get: \(2a^2 > a\) Now, subtract a from both sides: \(2a^2 - a > 0\) Second inequality: \(a^2 < 3a\) Subtracting 3a from both sides, we get: \(a^2 - 3a < 0\) Now, we will analyze both inequalities considering the \(x\) values.
05

Interval Based on the Inequalities

For the inequality \(2a^2 - a > 0\), we can factor out a: \(a(2a-1) > 0\) This inequality holds true for \(a < 0\) and \(a > \frac{1}{2}\). For the inequality \(a^2 - 3a < 0\), we can factor out a: \(a(a-3) < 0\) This inequality holds true for \(0 < a < 3\).
06

Find the Intersection of the Intervals

To determine the interval 'a' belongs to where the point falls inside the angle, we need to find the intersection of the intervals from both inequalities. The intersection of \(a < 0\) and \(0 < a < 3\) is empty, but the intersection of \(a > \frac{1}{2}\) and \(0 < a < 3\) is \(\frac{1}{2} < a < 3\). Therefore, the 'a' belongs to the interval \((\frac{1}{2}, 3)\), which corresponds to answer choice (b).

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Most popular questions from this chapter

Four points \(\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right),\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right),\left(\mathrm{x}_{3}, \mathrm{y}_{3}\right)\) and \(\left(\mathrm{x}_{4}, \mathrm{y}_{4}\right)\) are such that \({ }^{4} \sum_{\mathrm{i}=1}\left(\mathrm{x}_{\mathrm{i}}^{2}+\mathrm{y}_{\mathrm{i}}^{2}\right) \leq 2\left(\mathrm{x}_{1} \mathrm{x}_{3}+\mathrm{x}_{2} \mathrm{x}_{4}+\mathrm{y}_{1} \mathrm{y}_{2}+\mathrm{y}_{3} \mathrm{y}_{4}\right)\). Then these points are vertices of (a) Parallelogram (b) Rectangle (c) Square (d) Rhombus

Find the equation of line making a triangle of area \((50 / \sqrt{3})\) units with two axes and on which a perpendicular from origin makes an angle \((\pi / 6)\) with positive direction of \(\mathrm{x}\) -axis. (a) \(x+\sqrt{3 y}=10\) (b) \(x-y=10\) (c) \(\sqrt{3} x+y-5=0\) (d) \(\sqrt{3 x}+y-10=0\)

The equation of line passing through the point of intersection of the lines \(3 \mathrm{x}-2 \mathrm{y}=0\) and \(5 \mathrm{x}+\mathrm{y}-2=0\) and making the angle of measure \(\tan ^{-1}(-5)\) with the positive direction of \(\mathrm{x}\) -axis is \(\ldots \ldots\) (a) \(3 x-2 y=0\) (b) \(5 x+y-2=0\) (c) \(5 \mathrm{x}+\mathrm{y}=0\) (d) \(3 \mathrm{x}+2 \mathrm{y}+1=0\)

The length of side of an equilateral triangle is a. There is circle inscribed in a triangle. What is the area of a square inscribed in a circle ? (a) \(\left(\mathrm{a}^{2} / 3\right)\) (b) \(\left(\mathrm{a}^{2} / 6\right)\) (c) \(\left(\mathrm{a}^{2} / \sqrt{3}\right)\) (d) \(\left(a^{2} / \sqrt{2}\right)\)

The equation of a straight line that passes through the point \((-4,3)\) and is such that the portion of it between the axes is divided by the point in the ratio \(5: 3\) internally is (a) \(9 x-20 y+96=0\) (b) \(2 \mathrm{x}-\mathrm{y}+11=0\) (c) \(2 \mathrm{x}+\mathrm{y}+5=0\) (d) \(3 \mathrm{x}-2 \mathrm{y}+7=0\)
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