Origin In Obtuse Sector: Finding T Values

by Axel Sørensen 42 views

Hey everyone! Let's dive into an interesting problem in analytic geometry: finding the values of t for which the origin lies in the obtuse sector formed by two given lines. This problem combines concepts of linear equations, sectors formed by lines, and inequalities. We'll break it down step by step to make sure we understand every aspect.

Understanding the Problem

Before we jump into calculations, let's make sure we really get what the problem is asking. We've got two lines here:

  1. 4tx - 2ty + 3 - t = 0
  2. 3x + ty + t - 1 = 0

Our mission is to figure out for which values of t the origin (that's the point (0, 0)) sits inside the obtuse sector created by these lines. Remember, when two lines intersect, they form four sectors, two acute and two obtuse. The obtuse sectors are the wider ones, greater than 90 degrees.

Why is this important? Well, this kind of problem tests our understanding of how the coefficients of a linear equation relate to the line's position and orientation in the coordinate plane. It also touches on how to use inequalities to define regions in the plane.

Initial Thoughts and Approach

So, how do we tackle this? Here’s a general plan:

  1. Rewrite the equations: Express the equations in the standard form (Ax + By + C = 0) if they aren't already.
  2. Determine the conditions for the origin to lie in a sector: Think about what conditions must be met for the origin to be within a particular sector. This usually involves the signs of the expressions when (0, 0) is plugged into the line equations.
  3. Identify the obtuse sector condition: We need to figure out how to mathematically define the obtuse sector. This often involves considering the slopes of the lines and the angles between them.
  4. Solve the inequalities: Once we have our conditions set up, we'll need to solve the resulting inequalities to find the values of t.

Step-by-Step Solution

Let's get into the nitty-gritty now. We'll go through each step, making sure everything's clear.

1. Standard Form of Equations

Our equations are already in the standard form, which is awesome! We have:

  1. 4tx - 2ty + 3 - t = 0 (Line 1)
  2. 3x + ty + t - 1 = 0 (Line 2)

So, we can skip this step and move straight to the next one.

2. Condition for Origin to Lie in a Sector

This is where things get interesting. When the origin (0, 0) lies in a sector formed by the lines, it means that if we plug (0, 0) into the equations, the signs of the resulting expressions will tell us something important about the sector. Specifically, for the origin to lie in any sector, the product of the results when we substitute (0, 0) into the equations should have the same sign when considering one pair of sectors (acute or obtuse).

Let's plug (0, 0) into our equations:

  1. 4t(0) - 2t(0) + 3 - t = 3 - t
  2. 3(0) + t(0) + t - 1 = t - 1

Now, we consider the product of these results:

(3 - t)(*t* - 1)

For the origin to lie in a sector, this product must be either positive or negative. The specific sign will help us determine which sector (acute or obtuse) we are dealing with. The product must have the same sign for the origin to be on the same side of both lines, which is what defines a sector.

3. Identifying the Obtuse Sector Condition

This is the trickiest part. To figure out when the origin lies in the obtuse sector, we need an additional condition related to the slopes of the lines. Let's first find the slopes of the lines.

Finding Slopes

Rewrite the equations in slope-intercept form (y = mx + b), where m is the slope:

  1. Line 1: 2ty = 4tx + 3 - t => y = (2/t)x + (3 - t)/(2t) (provided t ≠ 0)
  2. Line 2: ty = -3x - t + 1 => y = (-3/t)x + (1 - t)/t (provided t ≠ 0)

So, the slopes are:

  • m1 = 2/t
  • m2 = -3/t

Condition for Obtuse Sector

For the origin to be in the obtuse sector, the lines should not be perpendicular, and the product of their slopes plus 1 should be negative. This ensures that the angle between the lines is greater than 90 degrees.

The condition is:

m1 m2 + 1 < 0

Plugging in the slopes:

(2/t)(-3/t) + 1 < 0

Simplifying:

-6/t^2 + 1 < 0

Multiply both sides by t^2 (assuming t^2 is positive, which it is since it's a square) and rearrange:

t^2 > 6

This gives us:

t < -√6 or t > √6

This condition ensures that the angle between the lines is obtuse.

4. Solving the Inequalities

Now we need to combine the condition for the origin to lie in a sector with the condition for the sector to be obtuse. We have two key inequalities:

  1. (3 - t)(*t* - 1) < 0 (Origin lies in a sector – corrected for obtuse sector)
  2. t < -√6 or t > √6 (Obtuse angle condition)

Let's solve the first inequality:

(3 - t)(*t* - 1) < 0

This inequality holds when the factors have opposite signs. We have two cases:

  • Case 1: 3 - t > 0 and t - 1 < 0
    • t < 3 and t < 1
    • This gives us t < 1
  • Case 2: 3 - t < 0 and t - 1 > 0
    • t > 3 and t > 1
    • This gives us t > 3

So, the solution for the first inequality is t < 1 or t > 3.

Combining the Inequalities

Now, we need to find the intersection of the solutions:

  1. t < 1 or t > 3
  2. t < -√6 or t > √6

Let's visualize this on a number line. We have the intervals:

  • (-∞, -√6)
  • (√6, ∞)
  • (-∞, 1)
  • (3, ∞)

√6 is approximately 2.45, so we can plot these on the number line.

The intersection gives us the following intervals for t:

  • t < -√6
  • t > 3

Final Answer

Therefore, the values of t for which the origin lies in the obtuse sector of the lines are t < -√6 or t > 3.

Visualizing the Solution

To really solidify our understanding, it's always a great idea to visualize the solution. Imagine plotting these lines for different values of t that satisfy our conditions (like t = -3 or t = 4). You'll see that the origin indeed falls within the wider, obtuse angle formed by the lines. If you pick a value of t that doesn't satisfy our conditions (like t = 2), you'll see the origin either falls in an acute sector or on one of the lines themselves.

Key Takeaways

  • Understanding Sectors: When two lines intersect, they divide the plane into four sectors. The obtuse sectors are the ones with angles greater than 90 degrees.
  • Origin and Line Position: The sign of the result when you plug the origin (0, 0) into the equation of a line tells you which side of the line the origin is on.
  • Slopes and Angles: The slopes of the lines determine the angles between them. The condition m1 m2 + 1 < 0 indicates an obtuse angle.
  • Combining Conditions: Solving problems like this often involves combining multiple conditions (inequalities) to narrow down the solution set.

Why This Matters

This problem isn't just a mathematical exercise; it's a fantastic example of how math can describe geometric relationships. These concepts are super useful in fields like computer graphics, where you need to determine if a point lies within a certain region, or in physics, when analyzing forces and angles. So, understanding how to solve these problems gives you a powerful set of tools for tackling real-world challenges.

I hope this breakdown helped clarify how to approach this kind of problem. Remember, the key is to break it down into smaller, manageable steps and to really understand the geometric principles at play. Keep practicing, and you'll become a pro at these in no time!