Sec 3 A-Math Coordinate Geometry & Circles topical questions

Find the midpoint and the gradient of the line joining (−1, 3) and (5, 11).

Find the equation of the line through (2, 7) that is parallel to y = 3x − 1.

Find the equation of the line through (2, 7) that is perpendicular to y = ½x.

Find the area of the triangle with vertices (0, 0), (6, 0) and (0, 4).

Write down the centre and radius of the circle x² + y² = 64.

The points A and B have coordinates (−2, 1) and (4, 9).

1. (a)Find the gradient of AB.[1]
2. (b)Find the midpoint of AB.[1]
3. (c)Find the equation of the perpendicular bisector of AB.[3]

Find the area of the quadrilateral with vertices (1, 1), (5, 2), (6, 6) and (2, 5), in that order.

The circle C has equation x² + y² + 8x − 2y − 8 = 0.

1. (a)Express the equation in the form (x − a)² + (y − b)² = r².[3]
2. (b)State the centre and radius of C.[2]
3. (c)State the coordinates of the point where C crosses the y-axis below the x-axis.[1]

Show that the point (7, −1) lies on the circle x² + y² − 6x + 2y − 6 = 0, and state the centre and radius of the circle.

The point (p, 5) lies on the circle x² + y² = 41.

1. (a)Form an equation in p.[2]
2. (b)Solve the equation.[2]
3. (c)Explain why there are two answers.[1]

Find the equation of the tangent to the circle x² + y² = 25 at the point (−3, 4).

1. (a)Find the gradient of the radius to (−3, 4).[2]
2. (b)Write down the gradient of the tangent.[1]
3. (c)Find the equation of the tangent, in the form ax + by = c.[3]

A circle has centre (2, −3) and passes through the point (6, 0).

1. (a)Find the radius of the circle.[2]
2. (b)Write the equation of the circle in centre–radius form.[2]
3. (c)Expand your answer into the general form.[2]

The points A, B and C have coordinates (1, 1), (5, 3) and (6, 1).

1. (a)Show that the triangle ABC is right-angled at B.[3]
2. (b)Find the area of triangle ABC by two different methods.[3]

Show that the line y = 2x + 5 is a tangent to the circle x² + y² = 5, and find the point of contact.

1. (a)Substitute the line into the circle and show that the resulting equation is (x + 2)² = 0.[3]
2. (b)Explain why this means the line is a tangent.[1]
3. (c)State the coordinates of the point of contact.[2]

The perpendicular bisector of the segment joining A(0, 4) and B(6, 0) is the locus of points equidistant from A and B. Find its equation.

1. (a)Find the midpoint of AB.[2]
2. (b)Find the gradient of the perpendicular bisector.[2]
3. (c)Find the equation of the perpendicular bisector, in the form ax + by = c.[2]