Parabola & Line Intersection: Find The Conditions!

Aug 3, 2025 by ADMIN 51 views

Unveiling the Mystery: When Parabolas and Lines Intersect Twice

Hey there, math enthusiasts! Ever wondered what happens when a parabola and a line decide to meet? It's not just a simple point of intersection sometimes; things can get a bit more interesting. Today, we're diving deep into a problem where a system of equations involving a parabola and a line has two solutions, and one of those solutions happens to be chilling at the parabola's vertex. Sounds intriguing, right? Let's break it down!

The Scenario: Tom's Two Solutions

Our friend Tom has stumbled upon a system of equations that's got him thinking. He's figured out that this system has two solutions. That's already a cool discovery! But here's the kicker: one of these solutions is hanging out at the vertex of the parabola. This adds a whole new layer to the problem, and it's our mission to figure out what conditions must be met for Tom's observation to be true. We need to explore the relationship between the parabola and the line to understand how they interact and create these two solutions, with one specifically at the vertex.

To really grasp this, let's meet the equations in question:

Equation 1: The Parabola

$(x - 3)^2 = y - 4$
Equation 2: The Line

$y = -x + b$

Now, let's roll up our sleeves and dissect these equations to uncover the secrets they hold.

Diving into the Parabola Equation

Let's start with the star of the show, the parabola. Equation 1, $(x - 3)^2 = y - 4$ , might look a bit intimidating at first, but it's actually in a super helpful form called vertex form. This form makes it incredibly easy to spot the parabola's vertex, which is a crucial point in our investigation.

Why is the vertex so important, you ask? Well, it's the turning point of the parabola, the place where it changes direction. It's either the lowest point (if the parabola opens upwards) or the highest point (if it opens downwards). In our case, the parabola opens upwards, making the vertex the minimum point.

So, how do we pinpoint the vertex from the equation? Remember the general vertex form of a parabola: $(x - h)^2 = a(y - k)$ , where $(h, k)$ is the vertex. Comparing this to our equation, $(x - 3)^2 = y - 4$ , we can clearly see that:

h = 3
k = 4

Therefore, the vertex of our parabola is at the point (3, 4). Mark this down; it's a key piece of information.

But there's more we can glean from this equation. The coefficient in front of the squared term (in this case, it's 1, since we can rewrite the equation as $1*(x - 3)^2 = y - 4$ ) tells us about the parabola's width and direction. Since it's positive, we know the parabola opens upwards. This is great for visualizing how the parabola will interact with the line.

Understanding the parabola's vertex and direction is like having a map for our problem. We know where the parabola's turning point is, and we know which way it's heading. Now, let's bring in the line and see how they interact.

Decoding the Line Equation

Now, let's turn our attention to the line, represented by Equation 2: $y = -x + b$ . This equation is in slope-intercept form, which is another fantastic form for quickly understanding a line's properties. Remember the general slope-intercept form: $y = mx + b$ , where 'm' is the slope and 'b' is the y-intercept.

In our equation, $y = -x + b$ , we can see that:

The slope (m) is -1. This means the line is sloping downwards as we move from left to right. For every one unit we move to the right on the x-axis, the line goes down one unit on the y-axis. Think of it as a gentle downhill ski slope.
The y-intercept is 'b'. This is the point where the line crosses the y-axis. It's the value of y when x is equal to 0.

The y-intercept, 'b', is the real mystery here. It's the only unknown in the line equation, and it's the key to unlocking the conditions that allow the line to intersect the parabola twice, with one intersection at the vertex. By changing the value of 'b', we can shift the line up or down, changing its points of intersection with the parabola.

Think of it this way: imagine the parabola as a fixed curve in the plane, and the line as a sliding ruler. By moving the ruler (changing 'b'), we can control where it intersects the parabola. Our goal is to find the sweet spot where the ruler intersects the parabola at the vertex and at one other point.

Now that we understand the individual characteristics of the parabola and the line, let's explore how they interact and what conditions on 'b' will lead to the scenario Tom described.

The Intersection Tango: When Line Meets Parabola

Here comes the exciting part: figuring out how the line and parabola interact! Remember, Tom observed that the system has two solutions, and one of them is at the parabola's vertex (3, 4). This is a crucial clue that will guide our steps.

To find the points of intersection, we need to solve the system of equations. This means finding the x and y values that satisfy both equations simultaneously. A classic way to do this is by substitution.

Since we know $y = -x + b$ (Equation 2), we can substitute this expression for 'y' into Equation 1: $(x - 3)^2 = y - 4$ . This gives us:

$(x - 3)^2 = (-x + b) - 4$

Now we have a single equation with only one variable, 'x'. Let's simplify and rearrange it into a more manageable form. Expanding the square and rearranging terms, we get:

$x^2 - 6x + 9 = -x + b - 4$

$x^2 - 5x + (13 - b) = 0$

Voila! We have a quadratic equation. Quadratic equations are known for having up to two real solutions, which perfectly aligns with Tom's observation of two intersection points. The solutions to this equation will give us the x-coordinates of the intersection points.

Now, remember that one of these intersection points is the vertex (3, 4). This means that x = 3 must be a solution to this quadratic equation. This is a game-changer! We can use this information to find the value of 'b'.

Substitute x = 3 into the quadratic equation:

$(3)^2 - 5(3) + (13 - b) = 0$

$9 - 15 + 13 - b = 0$

$7 - b = 0$

$b = 7$

We've cracked the code! We've found that b = 7 is a necessary condition for the line to intersect the parabola at the vertex. But is this enough to guarantee two solutions? Let's investigate further.

The Discriminant's Tale: Unveiling the Number of Solutions

We know that when b = 7, the line passes through the vertex of the parabola. But how do we ensure that there's another point of intersection? This is where the discriminant of a quadratic equation comes to the rescue.

The discriminant is a part of the quadratic formula that tells us about the nature of the roots (solutions) of a quadratic equation. For a quadratic equation in the form $ax^2 + bx + c = 0$ , the discriminant (often denoted as Δ) is given by:

$Δ = b^2 - 4ac$

Here's the crucial connection: The discriminant tells us how many real solutions the quadratic equation has:

If Δ > 0, the equation has two distinct real solutions.
If Δ = 0, the equation has one real solution (a repeated root).
If Δ < 0, the equation has no real solutions.

In our case, the quadratic equation is $x^2 - 5x + (13 - b) = 0$ . So, we have:

a = 1
b = -5 (not to be confused with the y-intercept 'b'!)
c = 13 - b

We want two distinct real solutions, so we need Δ > 0. Let's plug in the values and see what we get:

$Δ = (-5)^2 - 4(1)(13 - b)$

$Δ = 25 - 52 + 4b$

$Δ = 4b - 27$

For two distinct real solutions, we need:

$4b - 27 > 0$

$4b > 27$

$b > rac{27}{4}$

This is another vital piece of the puzzle! We've found that for the system to have two solutions, 'b' must be greater than 27/4 (which is 6.75). But remember, we already found that b = 7 is necessary for the line to pass through the vertex. Since 7 > 6.75, this condition is satisfied!

The Grand Finale: Conditions for Tom's Thinking to Be Correct

Phew! We've gone on quite the mathematical journey, dissecting parabolas, lines, and discriminants. Now, let's bring it all together and state the conditions that must be met for Tom's thinking to be correct.

Tom believes the system of equations has two solutions, one of which is located at the vertex of the parabola. For this to be true, we've established two key conditions:

The line must pass through the vertex: This means that when we substitute the vertex coordinates (3, 4) into the line equation $y = -x + b$ , the equation must hold true. This led us to the condition b = 7.
The system must have two distinct solutions: This is where the discriminant came into play. We found that the discriminant of the quadratic equation must be greater than zero, which led us to the condition $b > rac{27}{4}$ .

Since b = 7 satisfies both conditions (7 is indeed greater than 27/4), we can confidently say that Tom's thinking is correct when b = 7.

In conclusion, for the system of equations to have two solutions, with one at the parabola's vertex, the value of 'b' in the line equation must be equal to 7. This ensures that the line intersects the parabola at the vertex and at one other distinct point, creating the two solutions Tom observed.

So, there you have it! We've successfully unraveled the mystery of when parabolas and lines intersect twice, with a little help from our friend Tom and some powerful mathematical tools. Keep exploring, keep questioning, and keep the math magic alive!