A circle is tangent to the y-axis at y=3 and has one x-intercept at x=1.

(a) Determine the other x-intercept

(b) Deduce the equation of the circle.

My answer: EH? please help!

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- August 12th 2007, 11:38 PMkonnieHelp with a seemingly simple math problem!
A circle is tangent to the y-axis at y=3 and has one x-intercept at x=1.

(a) Determine the other x-intercept

(b) Deduce the equation of the circle.

My answer: EH? please help! - August 13th 2007, 12:29 AMCaptainBlack
- August 13th 2007, 02:57 AMticbol
The standard equation of a circle is

(x-h)^2 +(y-k)^2 = r^2

where

(h,k)is the center.

*"The circle is tangent to the y-axis at y=3."*

So a radius, r, is perpendicular to the y-axis at y=3.

Therefore, r = h, and k=3.

Draw the figure on paper.

Draw the horizontal radius h from (0,3) to (h,3) to see the relations.

Draw the radius h from (1,0) to (h,3).

By distance between two points,

h = sqrt[(1-h)^2 +(0-3)^2]

h^2 = (1 -2h +h^2) +9

2h = 10

h = 5

Draw the vertical radius to intersect the x-axis.

The horizontal chord, lying on the x-axis, of the circle is bisected by this vertical radius.

Since h=5, the intersection of this vertical radius and this horizontal chord is at (5,0).

Therefore, the (1,0) is 4 units to the left of (5,0).

Hence, the other end of the chord is at (9,0). <----the other x-intercept, answer.

Since (h,k) = (5,3), and r=h=5, the equation of the circle is

(x-5)^2 +(y-3)^2 = 25 ---------------answer.

-----------------

Check if the (9,0) satisfies the equation.

(9-5)^2 +(0-3)^2 =? 25

16 +9 =? 25

25 =? 25

Yes, so, OK. - August 13th 2007, 06:58 AMSoroban
Hello, konnie!

This is a variation of ticbol's solution.

. . Refer to Captain Black's diagram.

Quote:

A circle is tangent to the y-axis at and has an x-intercept at .

(a) Determine the other x-intercept

(b) Deduce the equation of the circle.

The center of the circle is: .

The y-intercept is: .

The x-intercept is: .

We find that . . and .

Since and are radii: .

. . and we have:. .

Hence, the center is and

**(b)**Therefore: .

Let

. .

**(a)**The other x-intercept is: .

- August 15th 2007, 12:01 AMkonnie
Thank you ^___^