In many applications, it is necessary to solve for the intersection of two curves. However, it's more complicated in general if we have y = ax^2 + bx + c (because of the "bx" term present) Solving the circle equation for \(x^2\) gives \(x^2 = 9 - y^2\). Looking at the distance from the vertex to the focus, \(p = 3 – 1 = 2\). Now set these two expressions for \(x^2\) equal to each other and solve. \[ - 21y^2 = - 125\nonumber\]Divide by -21 Continuing the situation from the last section, suppose stations C and D are located 200 km due south of stations A and B and 100 km apart. Note that if we divided by \(4p\), we would get a more familiar equation for the parabola, \(y = \dfrac{x^2}{4p}\). Plot both equations and see where they cross! \[\left( x - ( - 2) \right)^2 = 4( - 1)\left( y - 3 \right)\text{, or }\left( x + 2 \right)^2 = - 4\left( y - 3 \right)\nonumber\]. We can find the associated \(x\) values by substituting these \(y\)-values into either hyperbola equation. A radio telescope is 100 meters in diameter and 20 meters deep. The cannon ball flies through the air, following a parabola: y = 2 + 0.12x - 0.002x 2. It’s worth noting there is a second technique we could have used in the previous example, called elimination. A parabola with axis Y-axis is of the form [math]y = a{x}^2 + bx + c[/math] Let the points be [math](x_1, y_1), (x_2,y_2) [/math]and [math](x_3, y_3)[/math] First, ensure that the points are not collinear. Zoom out, then zoom in where they cross. \[1800 + \dfrac{9y^2}{16} = \dfrac{9(y + 200)^2}{91}\nonumber\]Divide by 9 Navigators would use other navigational techniques to decide between the two remaining locations. IB Examiner. Substituting those into \(y = 4x\) gives the corresponding \(y\) values. Write the standard conic equation for a parabola with vertex at the origin and focus at (0, -2). The method used is the method of substitution. The distance from the vertex to the focus, \(p\), is the focal length. In this example we solve the following pair of simultaneous equations: The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. From your earlier work with quadratics, you may already be able to identify the vertex as (1,2), but we’ll go ahead and put the parabola in the standard conic form. \[\dfrac{x^2}{2025} = 1 + \dfrac{y^2}{3600}\nonumber\]Multiply both sides by 2025 You should get something like this: By zooming in far enough we can find they cross at (25, 3.75), The "Standard Form" for the equation of a circle is (x-a)2 + (y-b)2 = r2. \[x = \pm \sqrt {\dfrac{16}{63}} = \pm 0.504\nonumber\]. While we studied parabolas earlier when we explored quadratics, at the time we did not discuss them as a conic section. In this example we solve the following pair of simultaneous equations: \[225 - 25y^2 + 4y^2 = 100\nonumber\]Combine like terms

For general parabolas, \[\dfrac{16x^2}{4} - \dfrac{x^2}{16} = 1\nonumber\]Multiply by 16 to get ), We can see they cross at about x = 0.7 and about x = 4.3. \[x \approx \pm 102.71\nonumber\], \[x^2 \approx 2025 + \dfrac{9( - 37.78)^2}{16}\nonumber\] You have already solved some examples of non-linear systems when you found the intersection of a parabola and line while studying quadratics, and when you found the intersection of a circle and line while studying circles. \[\begin{aligned} & x^2+y^2 = 5 \\ & x + y = 1 \end{aligned} \] Put the equation of the parabola \(y = 8{(x - 1)^2} + 2\) in standard conic form. When solving such simultaneous equations we're finding the coordinates (\(x\) and \(y\)) of the point(s) of intersection of a parabola and a line. The standard conic form of the equation is \[x^2 = 4py\nonumber\] Intersection of Line and Parabola. In our example, stations A and B are 150 kilometers apart and send a simultaneous radio signal to the ship. We need to determine the location of the focus, since that’s where the food should be placed. If a solar cooker has a parabolic dish 16 inches in diameter and 4 inches tall, where should the food be placed? & y = x^2 + 5x - 7 \\ We can solve this system of equations by substituting \(y = 2{x^2}\) into the hyperbola equation.

Using a similar process, we could find an equation of a parabola with vertex at the origin opening left or right. The curves intersect at (0.504, 2.016) and (-0.504, -2.016). \[x = \pm \sqrt {\dfrac{64}{21}} = \pm \dfrac{8}{\sqrt {21} }\nonumber\]. So, we will find the (x, y) coordinate pairs where a line crosses a parabola. When solving this type of pair of simultaneous equations we're finding the coordinates (\(x\) and \(y\)) of the point(s) of intersection of a circle and a line.