MHB [ASK] Equation of a Circle in the First Quadrant

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The center of circle L is located in the first quadrant on the line y = 2x, with the circle touching the Y-axis at (0, 6). The coordinates of the center can be determined as (3, 6) since the y-coordinate is derived from the tangential relationship to the Y-axis. The equation of the circle can be formulated as (x - 3)² + (y - 6)² = 3², leading to the simplified equation x² + y² - 6x - 12y + 36 = 0. The discussion also clarifies the correct use of "lies" versus "lays" in this context. The final equation represents the circle's properties accurately based on the given conditions.
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The center of circle L is located in the first quadrant and lays on the line y = 2x. If the circle L touches the Y-axis at (0,6), the equation of circle L is ...
a. $$x^2+y^2-3x-6y=0$$
b. $$x^2+y^2-12x-6y=0$$
c. $$x^2+y^2+6x+12y-108=0$$
d. $$x^2+y^2+12x+6y-72=0$$
e. $$x^2+y^2-6x-12y+36=0$$

Since the center (a, b) lays in the line y = 2x then b = 2a.
$$(x-a)^2+(y-b)^2=r^2$$
$$(0-a)^2+(6-b)^2=r^2$$
$$(-a)^2+(6-2a)^2=r^2$$
$$a^2+36-24a+4a^2=r^2$$
$$5a^2-24a+36=r^2$$
What should I do after this?
 
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circle center at $(x,2x)$

circle tangent to the y-axis at $(0,6) \implies x=3$

$(x-3)^2+(y-6)^2 = 3^2$

$x^2-6x+9+y^2-12y+36 =9$

$x^2+y^2-6x-12y+36=0$
 
skeeter said:
circle center at $(x,2x)$

circle tangent to the y-axis at $(0,6) \implies x=3$

How did you get x = 3 from (0, 6)?
 
Monoxdifly said:
How did you get x = 3 from (0, 6)?
The fact that the y-axis is tangent to the circle at (0, 6) means that the line y= 6 is a radius so the y coordinate of the center of the circle is 6. And since the center is at (x, 2x), y= 2x= 6 so x= 3.

(And the center of the circle "lies on the line y= 2x", not "lays".)
 
HallsofIvy said:
The fact that the y-axis is tangent to the circle at (0, 6) means that the line y= 6 is a radius so the y coordinate of the center of the circle is 6. And since the center is at (x, 2x), y= 2x= 6 so x= 3.

(And the center of the circle "lies on the line y= 2x", not "lays".)

Well, I admit I kinda suck at English. I usually use "lies" as "deceives" and "lays" as "is located". Thanks for the help, anyway. Your explanation is easy to understand.
 
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