Equation of a circle

In this question you must show detailed reasoning.

A circle has diameter PQ where P is (- 5, 1) and Q is (5, 1). The line x + 2 y = 12 meets the circle at A and B.

Find the exact length AB.

The curve C is defined, for 0 ⩽ t < 2 π, by the parametric equations

x = 4 k + k sin t, y = 2 + 4 cos t,

where k is a constant.

Find a cartesian equation for C. You do not need to simplify your answer.

You are given that C is a circle.

Determine the radius of C.

Find the possible coordinates for the centre of C.

A circle has centre C which lies on the x-axis, as shown in the diagram. The line y = x meets the circle at A and B. The midpoint of AB is M.

The equation of the circle is x² - 6x + y² + a = 0, where a is a constant.

In this question you must show detailed reasoning.

Show that the area of triangle ABC is

Find the value of a when the area of triangle ABC is zero.

Give a geometrical interpretation of the case in part (b)(i).

Give a geometrical interpretation of the case where a = 5.

The points P and Q have coordinates (2, - 5) and (3, 1) respectively.

Determine the equation of the circle that has PQ as a diameter. Give your answer in the form

x² + y² + ax + by + c = 0, where a, b and c are integers.

In this question you must show detailed reasoning.

Points A, B and C have coordinates (0, 6), (7, 5) and (6, −2) respectively.

Find an equation of the perpendicular bisector of AB.

Hence, or otherwise, find an equation of the circle that passes through points A, B and C.

A and B are fixed points in the x - y plane. The position vectors of A and B are a and b respectively.

State, with reference to points A and B, the geometrical significance of

the quantity | a - b |,

The circle P is the set of points with position vector p in the x - y plane which satisfy

State, in terms of a and b,

the position vector of the centre of P,

Enlarge

Find a cartesian equation of P.

A circle with centre C has equation x² + y² - 6x + 4y + 4 = 0.

Find

the coordinates of C,

the radius of the circle.

Determine the set of values of k for which the line y = kx - 3 does not intersect or touch the

circle.

A circle with centre C has equation x² +y² +8x-2y-7 = 0.

Find

the coordinates of C,

the radius of the circle.