Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 12andx ′ = rcos(α), y ′ = rsin(α) (1.13)Applying the addition formulas for the sine and cosine, the relationships in (1.12) can bewritten asx = rcosθcosα−rsinθsinαy = rsinθcosα+rcosθsinαand on substituting (1.13) in these equations we obtain the following relationships calledthe rotation equations:x = x ′ cosθ−y ′ sinθ(1.14)y = x ′ sinθ +y ′ cosθExample 1.11 Suppose that the axes of an xy-coordinate system are rotated through anangle of θ = 45 ◦ to obtain an x ′ y ′ -coordinate system. Find the equation of the curvein x ′ y ′ -coordinates.Solution .........x 2 −xy +y 2 −6 = 0If the rotation (1.14) are solved for x ′ and y ′ in terms of x and y, we obtain:x ′ = xcosθ +ysinθy ′ = −xsinθ+ycosθ(1.15)Example 1.12 Find the new coordinates of the point (2,4) if the coordinate axes are rotatedthrough an angle of θ = 30 ◦ .Solution .........Eliminating the Cross-Product TermIn Example 1.11, we were able to identify the curve x 2 −xy+y 2 −6 = 0 as an ellipse becausetherotationof axes eliminate the xy-term, thereby reducing the equation to a familiar form.This occurred because the new x ′ y ′ -axes were aligned with the axes of the ellipses. Thefollows theorem tells how to determine an appropriate rotation of axes to eliminate thecross-product term of a second-degree equation in x and y.Theorem 1.1 If the equationAx 2 +Bxy +Cy 2 +Dx+Ey +F = 0 (1.16)is such that B ≠ 0, and if an x ′ y ′ -coordinate system is obtained by rotating the xy-axesthrough an angle θ satisfyingcot2θ = A−C(1.17)Bthen, in x ′ y ′ -coordinates, Equation (1.16) will have the formA ′ x ′2 +C ′ y ′2 +D ′ x ′ +E ′ y ′ +F ′ = 0