What are the Points of Intersection of the Parabola y^2 8x and the Line x - 2y 6 0?

In this article, we will explore the process of finding the points of intersection between a parabola and a line. Specifically, we will delve into the calculation of the points of intersection between the parabola y^2 8x and the line x - 2y 6 0.

Steps to Find the Points of Intersection

To find the points of intersection between a parabola and a line, we can follow these systematic steps:

Rearrange the Line Equation: Express x in terms of y. Substitute into the Parabola Equation: Replace x in the parabola equation with the expression derived from the line equation. Solve the Resulting Quadratic Equation: Solve for y. Find Corresponding x Values: Substitute the y values back into the line equation to find the corresponding x values. State the Points of Intersection: Combine the x and y values to present the points of intersection.

Mathematical Steps for Finding the Intersection

Let's start with the given equations:

Parabola: y^2 8x Line: x - 2y 6 0

We begin by rearranging the line equation to express x in terms of y.

x 2y - 6

Next, we substitute this expression for x into the parabola equation y^2 8x:

y^2 8(2y - 6)

Simplifying the right side of the equation gives us:

y^2 16y - 48

Rearrange the equation to form a standard quadratic equation:

y^2 - 16y 48 0

Now, we solve the quadratic equation using the quadratic formula:

y frac{-b pm sqrt{b^2 - 4ac}}{2a}

Here, we have:

a 1 b -16 c 48

Substituting these values into the quadratic formula gives us:

y frac{16 pm sqrt{(-16)^2 - 4 cdot 1 cdot 48}}{2 cdot 1}

y frac{16 pm sqrt{256 - 192}}{2}

y frac{16 pm sqrt{64}}{2}

y frac{16 pm 8}{2}

This results in two solutions for y:

y_1 frac{24}{2} 12

y_2 frac{8}{2} 4

Find Corresponding x Values

We now substitute these y values back into the line equation x 2y - 6 to find the corresponding x values.

For y 12:

x 2(12) - 6 24 - 6 18

So, one point of intersection is (18, 12).

For y 4:

x 2(4) - 6 8 - 6 2

Thus, the other point of intersection is (2, 4).

Conclusion

Therefore, the points of intersection of the parabola and the line are:

(18, 12) (2, 4)

This process demonstrates a methodical approach to finding the intersection points of a parabola and a line, involving algebraic manipulation and the use of the quadratic formula. Understanding this process is fundamental in several branches of mathematics and can be extremely useful in various real-world applications.

Keywords: parabola intersection, line equation, quadratic formula