Difference between revisions of "2023 AMC 12B Problems/Problem 10"

Latest revision as of 11:54, 17 August 2024

Problem

In the $xy$ -plane, a circle of radius $4$ with center on the positive $x$ -axis is tangent to the $y$ -axis at the origin, and a circle with radius $10$ with center on the positive $y$ -axis is tangent to the $x$ -axis at the origin. What is the slope of the line passing through the two points at which these circles intersect?

$\textbf{(A)}\ \dfrac{2}{7} \qquad\textbf{(B)}\ \dfrac{3}{7} \qquad\textbf{(C)}\ \dfrac{2}{\sqrt{29}} \qquad\textbf{(D)}\ \dfrac{1}{\sqrt{29}} \qquad\textbf{(E)}\ \dfrac{2}{5}$

Solution 1

The center of the first circle is $(4,0)$ . The center of the second circle is $(0,10)$ . Thus, the slope of the line that passes through these two centers is $- \frac{10}{4} = - \frac{5}{2}$ .

Because this line is the perpendicular bisector of the line that passes through two intersecting points of two circles, the slope of the latter line is $\frac{-1}{- \frac{5}{2}} = \boxed{\textbf{(E) } \frac{2}{5}}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Solution 2 (Coordinate Geometry)

The first circle can be written as $(x-4)^2 + y^2 = 4^2$ we'll call this equation $(1)$ The second can we writen as $x^2 + (y-10)^2 = 10^2$ , we'll call this equation $(2)$

Expanding $(1)$ : $\begin{align*} x^2 -8x + 4^2 + y^2 &= 4^2 \\ x^2 - 8x + y^2 &= 0 \end{align*}$ Exapnding $(2)$ $\begin{align*} x^2 + y^2 -20y + 10^2 = 10^2\\ x^2 + y^2 - 20y = 0 \end{align*}$

Now we can set the equations equal to eachother: $\begin{align*} x^2 - 8x + y^2 &= x^2 + y^2 - 20y \\ \frac{8}{20}x &= y \\ \frac{2}{5}x &= y \end{align*}$ This is in slope intercept form therefore the slope is $\boxed{\textbf{(E) } \frac{2}{5}}$ .

~luckuso

Video Solution 1 by OmegaLearn

https://youtu.be/IUB6r1iNgpU

So um you can write both circles in polar form: The first circle with radius 4 um *long pause can be rewritten as r=8cosθ. And the other circle can be written as r=20sinθ. At the point of intersection, both radii will be equal. We can then do 8cosθ=20sinθ *clears throat. Rearrange the equation to solve for tanθ so tanθ = 2/5. Tanθ = y/x which yields the slope, so the slope is 2/5 $\boxed{\textbf{(E) } \frac{2}{5}}$ moomoo. Thechickenonfire is the moomoo cow and you should milk the cow of knowledge.

-By Elite_Trash777

Video Solution

https://youtu.be/sjJMWtL_CEY

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

@@ Line 11: / Line 11: @@
 Because this line is the perpendicular bisector of the line that passes through two intersecting points of two circles, the slope of the latter line is <math>\frac{-1}{- \frac{5}{2}} = \boxed{\textbf{(E) } \frac{2}{5}}</math>.
 ~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
@@ Line 37: / Line 38: @@
 \end{align*}</cmath>
 This is in slope intercept form therefore the slope is <math>\boxed{\textbf{(E) } \frac{2}{5}}</math>.
+[[Image:Amc12B_2023_P10.PNG|thumb|center|600px]]
 ~[https://artofproblemsolving.com/wiki/index.php/User:Cyantist luckuso]
 ==Video Solution 1 by OmegaLearn==
 https://youtu.be/IUB6r1iNgpU
+So um you can write both circles in polar form: The first circle with radius 4 um *long pause can be rewritten as r=8cosθ. And  the other circle can be written as r=20sinθ. At the point of intersection, both radii will be equal. We can then do 8cosθ=20sinθ *clears throat. Rearrange the equation to solve for tanθ so tanθ = 2/5. Tanθ = y/x which yields the slope, so the slope is 2/5 <math>\boxed{\textbf{(E) } \frac{2}{5}}</math> moomoo. Thechickenonfire is the moomoo cow and you should milk the cow of knowledge.
+-By Elite_Trash777
 ==Video Solution==

2023 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
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