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# a circle is tangent to the y-axis at y=3 and has one x intercept at x=1 determine the other x intercept

deduce the equation of the circle

When a circle is tangent to a line, the line is perpendicular to the radius at the point of tangency. So since this circle is tangent to the y-axis at y=3, the y-coordinate of the center of the circle must also be 3. Let cx be the x-coordinate of the center of this circle. cx is also the radius, since the distance between the point of tangency (0,3) and the center of the circle (cx,3) is cx.So by the Pythagorean Theorem we have

cx2 = (cx - 1)+ (3 - 0)2.

Hence 0 = -2cx + 10. So cx = 5. It follows that the equation of the circle is

(x - 5)2 + (y - 3)2 = 52.

Rather than using the equation of the circle to find the other x-intercept by brute force, let us use the fact that both the circle and the line it intersects (the x-axis) have mirror symmetry about the line x = cx. So any feature (such as an intersection) of the two which appears at (x, y) must also appear at (cx+ (c- x), y) = (10 - x, y). So since there is an x-intercept of x=1, i.e. appearing at (1, 0), there must be one at (9, 0), i.e. an x-intercept of x=9.

Ok, this one is a bit tricky to discuss without drawing a figure so bear with me.

The equation of a circle is (x -x0)2 + (y - y0)2 = r2 where (x0, y0) is the location of the circles center. We have three unknowns to find including the radius. Draw a bunch of circles tangent to the y-axis. Perpendicular to the point of tangency is a radius so that tells us that the height of the center is at the point of tangency so y0 = 3. So the equation is now (x -x0)2 + (y - 3)2 = r2

Lets use another piece of information: at x = 0, y = 3 since the circle touches the y-axis.

(0 -x0)2 + (3 - 3)2 = ror

x02 = r2

So once we know the x-coordinate of the center, we'll know r.

Lets use the known x-intercept and plug the numbers into the equation for the circle.

(1 -x0)2 + (0 - 3)2 = r2

(1 -x0)2 + 9 = r2

In the equation we just derived, if we replace r2 with x02 using the last formula we derived, we'll get x0 = 5 and r = 5. So the equation is

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

Plug the given point of tangency and the x-intercept into the formula as a check. Both sides should be equal to 25. Then, replace y with 0 and find the second x-intercept (x = 9)