Analytical geometry by Vaisman L.

By Vaisman L.

This quantity discusses the classical topics of Euclidean, affine and projective geometry in and 3 dimensions, together with the type of conics and quadrics, and geometric alterations. those matters are vital either for the mathematical grounding of the coed and for purposes to numerous different matters. they are studied within the first 12 months or as a moment path in geometry. the cloth is gifted in a geometrical method, and it goals to boost the geometric instinct and taking into consideration the scholar, in addition to his skill to appreciate and provides mathematical proofs. Linear algebra isn't a prerequisite, and is saved to a naked minimal. The booklet encompasses a few methodological novelties, and numerous workouts and issues of strategies. It additionally has an appendix concerning the use of the pc programme MAPLEV in fixing difficulties of analytical and projective geometry, with examples.

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1] (ii) Find the equation of the line l2 which is parallel to l1 and passes through the point (1, -2). [3] 2. Describe fully the curve whose equation is x 2 + y 2 = 4 . [2] 3. The coordinates of two points are A (-1, -3) and B (5, 7). Calculate the equation of the perpendicular bisector of AB. [4] 4. Show that the line y = 3x – 10 is a tangent to the circle x 2 + y 2 = 10 . [4] 5. The line y = 2 x − 3 meets the x-axis at the point P, and the line 3 y + 4 x = 8 meets the x-axis at the point Q. The two lines intersect at the point R.

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.

I) Find the coordinates of R. [4] (ii) Find the area of triangle PQR. [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.

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