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

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SUMMARY

The discussion focuses on solving the equation 4c=1+4r^2, derived from a question in the HSC Advanced Maths exam in Australia. Participants emphasize the need to find the intersection points between a circle and a parabola, leading to a quartic equation in x that can be reduced to a quadratic in x². The solution requires analyzing the discriminant of the quadratic and determining conditions for the sum of its roots.

PREREQUISITES
  • Understanding of quadratic equations and their discriminants
  • Familiarity with the concepts of circles and parabolas in coordinate geometry
  • Knowledge of solving quartic equations
  • Basic skills in symbolic manipulation and algebraic expressions
NEXT STEPS
  • Study the properties of quadratic equations and their discriminants
  • Learn about the geometric interpretations of circles and parabolas
  • Explore methods for solving quartic equations
  • Investigate the conditions for the sum of roots in polynomial equations
USEFUL FOR

Students preparing for advanced mathematics exams, educators teaching coordinate geometry, and anyone interested in mastering polynomial equations and their applications in geometry.

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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.
 
Last edited:

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