Solve 4c=1+4r^2: HSC Advanced Maths Exam Ques.

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The discussion revolves around solving the equation 4c=1+4r^2 from the HSC Advanced Maths exam in Australia. Participants express confusion about how to approach the problem, particularly in finding the intersection points between a circle and a parabola. It is noted that the equation is a quartic in x, which can be reduced to a quadratic in x^2. The discussion emphasizes the importance of using the condition of symmetry to determine the discriminant of the quadratic. Ultimately, the focus is on deriving conditions for the roots of the quadratic to solve the problem effectively.
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Homework Statement


http://www.mediafire.com/?rhnyop6erds34db

This was a question from the HSC Advanced maths exam in Australia, the second easiest maths course.

Homework Equations



given*

The Attempt at a Solution


I have no idea ):
 
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An idea to start is to solve the system for x and y, even though they will be in symbolic form. Notice, that in i, what you are asked to show is not expressed with x or y.
 
hscguy said:

Homework Statement


http://www.mediafire.com/?rhnyop6erds34db

This was a question from the HSC Advanced maths exam in Australia, the second easiest maths course.

Homework Equations



given*

The Attempt at a Solution


I have no idea ):

Find the equation that gives the solution set for the intersection points between the circle and the parabola. This is a quartic in x, reducible to a quadratic in x2. Now use the stipulation given ("located symmetrically") to deduce a condition that the discriminant of that quadratic must satisfy.

For the second part, figure out what condition the sum of the roots of that quadratic has to satisfy.
 
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