Origin In Obtuse Sector: Finding T Values
Hey guys! Today, we're diving into a cool problem from analytic geometry: finding the values of 't' that make the origin lie within the obtuse sector formed by two lines. This might sound a bit intimidating at first, but don't worry, we'll break it down step by step. We're talking about lines defined by equations with a parameter 't', and we need to figure out when the origin (that's the point (0,0) on our coordinate plane) sits inside the wider angle created where these lines intersect. Think of it like finding the perfect spot for a picnic blanket in the shade of two trees – except our trees are lines, and the shade is an obtuse angle!
Understanding the Problem
Before we jump into calculations, let's make sure we really get what the question is asking. We've got two lines:
4tx - 2ty + 3 - t = 03x + ty + t - 1 = 0
These aren't just any lines; they change depending on the value of 't'. Our mission is to find all the 't' values that make the origin hang out in the obtuse sector formed by these lines. Remember, an obtuse angle is any angle greater than 90 degrees but less than 180 degrees. So, imagine these two lines slicing up the plane like pizza slices. We want the origin to be in the slice that's wider than a right angle.
The key here is understanding how the position of the origin relative to a line is determined by the sign of the expression you get when you plug the origin's coordinates (0, 0) into the line's equation. If the signs are the same for both lines, the origin is in the acute sector. If the signs are different, the origin is in the obtuse sector. This is the core concept we'll use to solve this problem. We need to find the values of t for which plugging (0, 0) into both equations results in expressions with opposite signs. This ensures that the origin lies within the obtuse angle formed by the two lines.
To kick things off, we'll plug the origin (0, 0) into each line equation. This will give us two expressions in terms of 't'. Then, we'll analyze the signs of these expressions. We're looking for the values of 't' that make these signs different – one positive and one negative. This condition will translate into an inequality involving 't', which we can then solve to find the range of 't' values that satisfy our condition. Remember, the beauty of analytic geometry lies in translating geometric conditions (like the origin being in an obtuse sector) into algebraic equations and inequalities, which we can then solve using our familiar mathematical tools. So, let's roll up our sleeves and get those 't' values!
Plugging in the Origin
Okay, let's get our hands dirty and plug the origin (0, 0) into our line equations. This is a crucial step because it simplifies the problem and gives us something concrete to work with. Remember, we're trying to figure out the relationship between the origin and the lines, and this is the direct way to do it.
For the first line, 4tx - 2ty + 3 - t = 0, if we substitute x = 0 and y = 0, we get:
4t(0) - 2t(0) + 3 - t = 0 + 0 + 3 - t = 3 - t
So, the expression we get for the first line is 3 - t. This is a simple linear expression, and its sign will depend on the value of 't'. If 't' is less than 3, this expression is positive. If 't' is greater than 3, it's negative. And if 't' is exactly 3, it's zero. But we're interested in the signs, so we'll keep that in mind.
Now, let's do the same for the second line, 3x + ty + t - 1 = 0. Plugging in x = 0 and y = 0, we have:
3(0) + t(0) + t - 1 = 0 + 0 + t - 1 = t - 1
For the second line, our expression is t - 1. Again, this is linear. If 't' is less than 1, this expression is negative. If 't' is greater than 1, it's positive. And if 't' is 1, it's zero.
So, we've got our two expressions: 3 - t from the first line and t - 1 from the second line. These are the key players in our quest. Remember, for the origin to be in the obtuse sector, these two expressions need to have opposite signs. That means one needs to be positive while the other is negative. Now, we need to figure out the 't' values that make this happen. This is where we'll use inequalities to express the conditions for opposite signs and solve for 't'. It's like setting up a sign-matching game for 't', and the rules are determined by whether the expressions are positive or negative. Let's move on to setting up those inequalities!
Setting Up the Inequalities
Alright, guys, now comes the crucial part where we translate the
