Intersecting lines,circles,and parabolas

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Homework Help Overview

The discussion revolves around finding the intersection points of the graphs defined by the equations y=2x and x^2+y^2=1, which represents a line and a circle, respectively. There is also a mention of a parabola, leading to questions about the methods for determining intersection points in general.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss substituting one equation into another to find intersection points, with some suggesting solving for x and y directly. There are questions about whether similar methods apply to parabolas and the nature of the equations involved.

Discussion Status

Some participants have provided guidance on how to approach solving the equations, while others have clarified the nature of the equations involved. There is an acknowledgment of the need to focus on solving rather than the geometric interpretation.

Contextual Notes

There is some confusion regarding the identification of the second equation as a parabola, which has been corrected to clarify it is a circle. Participants express a desire for simple explanations and clarity in the methods used.

Sumaya
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Homework Statement



find the points in which the graphs intersect

Homework Equations



y=2x, x^2+y^2=1

The Attempt at a Solution



the center point of the circle is (0,0)
and the radius = 1
and i collect some points to draw the line equation
(-1,-2) (0,0) (1,2) (2,4) etc ..

but i don't know how to get the intersect points ..
and also for the parabola equation ... are they have same way to know the intersect points ...
is there a rule or what ??
please explain to me in simple english words..

thanx a lot ...
 
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Just solve for x and y. Plug y = 2x into
x2+y2=1
to get
x2+(2x)2=1

Solve for x (you should get two solutions), and plug both into
y = 2x
to get the corresponding solutions for y.

Sumaya said:
and also for the parabola equation ... are they have same way to know the intersect points ...
is there a rule or what ??
The 2nd equation is an equation of a circle, not a parabola.
 
In simple English (nicht Deutsch, warum?), don't worry about the geometry, solve the equations!

If the problem were to find the intersection of [itex]x^2+ y^2= 1[/itex] and [itex]y= x^2[/itex], the first is a circle and the second a parabola. That's nice to know (it tells us we can expect to find two points of intersection) but not necessary to the solution. Since [itex]y= x^2[/itex] we can replace [itex]x^2[/itex] by y in the first equation: [itex]y+ y^2= 1[/itex] or [itex]y^2+ y- 1= 0[/itex]. That's a quadratic equation and we can either complete the square or use the quadratic formula to solve for y. Once we have found y, x is a square root.

(The quadratic equation has two roots, of course, and you might think that since each has two roots, there would be 4 (x, y) combinations. But one of the (real) roots to the quadratic is negative. That gives only imaginary roots and coordinates of points in a graph must be real. Only y> 0 gives the two points of intersection.)
 
eumyang said:
Just solve for x and y. Plug y = 2x into
x2+y2=1
to get
x2+(2x)2=1

Solve for x (you should get two solutions), and plug both into
y = 2x
to get the corresponding solutions for y.


The 2nd equation is an equation of a circle, not a parabola.

thanx a lot ...
 
HallsofIvy said:
In simple English (nicht Deutsch, warum?), don't worry about the geometry, solve the equations!

you are right ..

and i undestand how to solve the equation ...

thanx a lot ..
 

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