# [ASK] A Circle Which Touches the X-Axis at 1 Point

• MHB
• Monoxdifly
In summary, the given equation represents a circle that touches the X-axis at one point. The center of the circle is located at either (3, -4) or (-3, -4), with the coefficient of y in the equation indicating the ordinate of the center. The value of p can be determined by noticing that the equation can be rewritten as a perfect square, and thus only one of the given options, namely (3, -4), satisfies the equation.
Monoxdifly
MHB
The circle $$\displaystyle x^2+y^2+px+8y+9=0$$ touches the X-axis at one point. The center of that circle is ...
a. (3, -4)
b. (6, -4)
c. (6, -8)
d. (-6, -4)
e. (-6, -8)

I already eliminated option c and e since based on the coefficient of y in the equation, the ordinate of the center must be -4. However, I don't know how to determine the abscissa since we need to determine the value of p first, in which we need to substitute the value of x and y while the only info I have is y = 0. How should I do this?

circle touches the x-axis at one point $\implies x^2+px+9$ is a perfect square $\implies p$ is either +6 or -6 $\implies x$ is either -3 or +3,

choice (a) seems to be the only plausible fit ...

$x^2-6x+9 +y^2+8y+16=16$

$(x-3)^2 +(y+4)^2 =4^2$

Ah, thanks skeeter! That was faster than I thought...

## 1. What is a circle that touches the X-axis at 1 point?

A circle that touches the X-axis at 1 point is known as a tangent circle. This means that the circle only intersects the X-axis at one point, creating a right angle with the X-axis at that point.

## 2. How is the center of the circle determined?

The center of the circle is determined by finding the midpoint between the point of tangency on the X-axis and the center of the X-axis. This point will be the center of the circle.

## 3. What is the equation for a circle that touches the X-axis at 1 point?

The equation for a circle that touches the X-axis at 1 point is (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The value of h can be found by taking the X-coordinate of the point of tangency, and the value of k can be found by taking the Y-coordinate of the center of the X-axis.

## 4. How does the tangent circle relate to the X-axis?

The tangent circle is perpendicular to the X-axis at the point of tangency. This means that the line tangent to the circle at the point of tangency is perpendicular to the X-axis, creating a right angle.

## 5. Can a circle touch the X-axis at more than 1 point?

No, a circle can only touch the X-axis at 1 point. If a circle intersects the X-axis at more than 1 point, it is not a tangent circle. A tangent circle only has 1 point of tangency with the X-axis, creating a right angle.

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