Circles：练习题

A circle has a radius of 7. What is the area of the circle?

A circle has a circumference of \(36\pi\). What is the length of an arc of this circle that is subtended by a central angle of \(120^\circ\)?

In the xy-plane, the graph of the equation \(x^2 + y^2 + 8x - 12y = c\) is a circle, where \(c\) is a constant. If the circle is tangent to the x-axis, what is the value of \(c\)?

The equation of a circle in the \(xy\)-plane is given by \(x^2 + y^2 - 10x + 12y + 57 = 0\). What is the radius of this circle?

Point \(P\) is outside a circle with center \(O\). Segments \(PA\) and \(PB\) are tangent to the circle at points \(A\) and \(B\), respectively. If the measure of the major arc AB is \(240^\circ\), what is the measure of \(\angle APB\), in degrees?

In the \(xy\)-plane, a circle is defined by the equation \(x^2 + y^2 - 4x + 10y = 7\). What is the radius of this circle?

A circle in the \( xy \)-plane has its center at \( (3, -2) \). A line segment with endpoints \( P(7, 1) \) and \( Q(k, 5) \) is tangent to the circle at point \( P \).

Part A: Find the radius of the circle by calculating the distance between the center and point \( P \).
Part B: Determine the slope of the radius connecting the center to the tangent point \( P \).
Part C: Using the fact that a tangent line is perpendicular to the radius at the point of tangency, find the slope of line segment \( PQ \).
Part D: Use the slope found in Part C to solve for the value of the constant \( k \).