Difference between revisions of "1988 AIME Problems/Problem 14"

Revision as of 01:16, 17 November 2007

Let $C$ be the graph of $xy = 1$ , and denote by $C^*$ the reflection of $C$ in the line $y = 2x$ . Let the equation of $C^*$ be written in the form

$12x^2 + bxy + cy^2 + d = 0.$

Find the product $bc$ .

$bc=84$

We need to find a formula for $P^*$ if $P=(x,y)$ .

With some algebra, we find this to be:

$P^*=(\frac{-3x+4y}{5},\frac{4x+3y}{5})$ .

Substituting in, we get:

$\frac{(-3x+4y)(4x+3y)}{25}=1$

Expanding,

$12x^2-7xy-12y^2+25=0$ .

Thus, $bc=(-7)(-12)=84.$

@@ Line 7: / Line 7: @@
 == Solution ==
-{{solution}}
+<math>bc=84</math>
+We need to find a formula for <math>P^*</math> if <math>P=(x,y)</math>.
+With some algebra, we find this to be:
+<math>P^*=(\frac{-3x+4y}{5},\frac{4x+3y}{5})</math>.
+Substituting in, we get:
+<math>\frac{(-3x+4y)(4x+3y)}{25}=1</math>
+Expanding,
+<math>12x^2-7xy-12y^2+25=0</math>.
+Thus, <math>bc=(-7)(-12)=84.</math>
 == See also ==

1988 AIME (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions