By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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**Extra resources for Affine Algebraic Geometry**

**Example text**

3] (vi) Calculate the area of the quadrilateral PQRS. 8. AB is the diameter of a circle. A is (1, 3) and B is (7, -1). [2] (i) Find the coordinates of the centre C of the circle. [2] (ii) Find the radius of the circle. [2] (iii) Find the equation of the circle. (iv) The line y + 5x = 8 cuts the circle at A and again at a second point D. Calculate the coordinates of D. [4] [3] (v) Prove that the line AB is perpendicular to the line CD.

3] 6. The equation of a circle is x 2 + y 2 − 4 x + 2 y = 15 (i) Find the coordinates of the centre C of the circle, and the radius of the circle. (ii) Show that the point P (4, -5) lies on the circle. (iii) Find the equation of the tangent to the circle at the point P. [3] [1] [4] 7. The coordinates of four points are P (-2, -1), Q (6, 3), R (9, 2) and S (1, -2). [4] (i) Calculate the gradients of the lines PQ, QR, RS and SP. [1] (ii) What name is given to the quadrilateral PQRS? [2] (iii) Calculate the length SR.

Iv) Show that the equation of SR is 2y = x – 5 and find the equation of the line L through Q perpendicular to SR. [5] [3] (v) Calculate the coordinates of the point T where the line L meets SR. [3] (vi) Calculate the area of the quadrilateral PQRS. 8. AB is the diameter of a circle. A is (1, 3) and B is (7, -1). [2] (i) Find the coordinates of the centre C of the circle. [2] (ii) Find the radius of the circle. [2] (iii) Find the equation of the circle. (iv) The line y + 5x = 8 cuts the circle at A and again at a second point D.