Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 14: Problem 1203

If \(\left(a, a^{2}\right)\) lies inside the angle between the lines \(y=(x / 2)\), \(\mathrm{x}>0\) and \(\mathrm{y}=3 \mathrm{x}, \mathrm{x}>0\), then \(\mathrm{a} \in \ldots \ldots\) (a) \([-3,-(1 / 2)]\) (b) \((3, \infty)\) (c) \([-(1 / 2), 3]\) (d) \([0,(1 / 2)]\)

Short Answer

Expert verified
The possible values of \(a\) are in the open interval \((\frac{1}{2}, 3)\), so the correct answer is (c) \([- \left(\frac{1}{2}\right), 3]\).

Step by step solution

01

Determine the relationship between the point and the line y = (x / 2)

Since the point \((a, a^2)\) lies inside the angle formed by the lines, it should satisfy the inequality formed by the line \(y =\left(\frac{x}{2}\right)\). Considering only the positive values of x, we want to find if this inequality is greater or lesser for y. First, let's plug in the coordinates of the point: \(a^2 > \frac{a}{2}\) (because x > 0) Now, let's multiply both sides by 2, to get rid of the fraction: \(2 a^2 > a \)
02

Determine the relationship between the point and the line y = 3x

Now we will do the same for the line \(y = 3x\). Since the given point \((a, a^2)\) lies inside the angle formed by the lines, it should satisfy the inequality formed by the line \(y = 3x\). Considering only the positive values of x, we want the inequality for y to be lesser. Plug in the coordinates of the point: \(a^2 < 3a\)
03

Solve the inequalities

Now we have two inequalities that we need to solve: 1. \(2 a^2 > a\) 2. \(a^2 < 3a\) Let's solve the inequalities one by one. For inequality 1: \(2a^2 > a\) Subtract a from both sides: \(2a^2 - a > 0\) Factor out a: \(a(2a - 1) > 0\) Since \(a \neq 0\) (because a has to be positive), we can determine that: \(a > \frac{1}{2}\) For inequality 2: \(a^2 < 3a\) Subtract 3a from both sides: \(a^2 - 3a < 0\) Factor a: \(a(a - 3) < 0\) Since \(a \neq 0\) (because a has to be positive), we can determine that: \(a < 3\)
04

Combine the inequalities

Now that we have solved both inequalities individually, let's combine them to find the range of a: 1. \(a > \frac{1}{2}\) 2. \(a < 3\) These inequalities can be combined into one compound inequality: \(\frac{1}{2} < a < 3\) This means that the possible values of \(a\) are in the open interval \((\frac{1}{2}, 3)\) which can also be written as \([- \left(\frac{1}{2}\right), 3]\) in the given options. The correct answer is (c) \([- \left(\frac{1}{2}\right), 3]\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A rod having length \(2 \mathrm{c}\) moves along two perpendicular lines, then the locus of the midpoint of the rod is \(\ldots \ldots\) (a) \(x^{2}-y^{2}=c^{2}\) (b) \(x^{2}+y^{2}=c^{2}\) (c) \(x^{2}+y^{2}=2 c^{2}\) (d) None of these

Three straight lines \(2 \mathrm{x}+11 \mathrm{y}-5=0,4 \mathrm{x}-3 \mathrm{y}-2=0\) and \(24 x+7 y-20=0\) (a) form a triangle (b) are only concurrent (c) are concurrent with one line bisecting the angle between the other two. (d) none of these

If for \(a+b+c \neq 0\), the lines \(a x+b y+c=0, b x+c y+a=0\) and \(\mathrm{cx}+\mathrm{ay}+\mathrm{b}=0\) are concurrent, then \(\ldots \ldots \ldots\) (a) \(a b+b e+c a=0\) (b) \((\mathrm{a} / \mathrm{b})+(\mathrm{b} / \mathrm{c})+(\mathrm{c} / \mathrm{a})=1\) (c) \(a=b\) (d) \(a=b=c\)

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) \([(\sqrt{3}-1) / 2]\)

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) \(\left(a^{2} / \sqrt{2}\right)\)
See all solutions

Recommended explanations on English Textbooks

English Language Study

Read Explanation

Key Concepts in Language and Linguistics

Read Explanation

Blog

Read Explanation

Synthesis Essay

Read Explanation

Global English

Read Explanation

Single Paragraph Essay

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free