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AMC 10 Geometry Math Olympiad USA Math Olympiad

Graph Coordinates | AMC 10A, 2015 | Question 12

Try this beautiful Problem on Graph Coordinates from co-ordinate geometry from AMC 10A, 2015. You may use sequential hints to solve the problem.

Try this beautiful Problem on Graph Coordinates from coordinate geometry from AMC 10A, 2015.

Graph Coordinates – AMC-10A, 2015- Problem 12


Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^{2}+x^{4}=2 x^{2} y+1 .$ What is $|a-b| ?$

,

  • $0$
  • $1$
  • $2$
  • $3$
  • \(4\)

Key Concepts


Co-ordinate geometry

graph

Distance Formula

Check the Answer


Answer: $2$

AMC-10A (2015) Problem 12

Pre College Mathematics

Try with Hints


The given points are $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ which are satisfying the equation $y^{2}+x^{4}=2 x^{2} y+1$.

So we can write $y^{2}+\sqrt{\pi}^{4}=2 \sqrt{\pi}^{2} y+1$

Can you now finish the problem ……….

Therefore

$y^{2}+\pi^{2}=2 \pi y+1$
$y^{2}-2 \pi y+\pi^{2}=1$
$(y-\pi)^{2}=1$
$y-\pi=\pm 1$
$y=\pi+1$
$y=\pi-1$

can you finish the problem……..

There are only two solutions to the equation, so one of them is the value of $a$ and the other is $b$. our requirement is $|a-b|$ so between a and b which is greater is not importent…………

So, $|(\pi+1)-(\pi-1)|=2$

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