Question

in the xy - plane, a circle intersects the line represented by ( y = 22 ) at exactly one point, and the circle intersects the line represented by ( x = 21 ) at exactly one point. an equation of this circle can be written in the form ( (x - h)^2+(y - k)^2 = 49 ), where ( h ) and ( k ) are constants. what is a possible value of ( h )?

Explanation:

Step1: Analyze the property of the circle and the lines

A circle intersecting a line at exactly one point means the line is tangent to the circle.
If the circle \((x - h)^{2}+(y - k)^{2}=r^{2}\) (here \(r^{2} = 49\), so \(r = 7\)) is tangent to the line \(x=21\), the distance from the center \((h,k)\) of the circle to the line \(x = 21\) is equal to the radius of the circle.
The distance \(d\) from a point \((x_0,y_0)\) to the line \(x=a\) is given by \(d=\vert x_0 - a\vert\). Here \(x_0=h\), \(y_0 = k\), and \(a = 21\), and \(d=r = 7\).

Step2: Solve for \(h\)

Using the distance formula \(\vert h-21\vert=7\).
We have two cases:
Case 1: \(h-21=7\), then \(h=21 + 7=28\).
Case 2: \(h-21=-7\), then \(h=21-7 = 14\).

Answer:

\(14\) (or \(28\))