Chapter 1 Topics in Analytic Geometry
Chapter 1 Topics in Analytic Geometry 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
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 13Remark• It is always possible to satisfy (1.17) with an angle θ in the range 0 < θ < π/2. Weshall always use such a value of θ.• The values of sinθ and cosθ needed for the rotation equations may be obtained byfirst calculating cos2θ and then computing sinθ and cosθ from the identitiessinθ =√1−cos2θ2and cosθ =√1+cos2θ2Example 1.13 Discuss and sketch the graph of the equation xy = 1.Solution .........Example 1.14 Identify and sketch the curve41x 2 −24xy +34y 2 −25 = 0Solution .........
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MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 13Remark• It is always possible to satisfy (1.17) with an angle θ <strong>in</strong> the range 0 < θ < π/2. Weshall always use such a value of θ.• The values of s<strong>in</strong>θ and cosθ needed for the rotation equations may be obta<strong>in</strong>ed byfirst calculat<strong>in</strong>g cos2θ and then comput<strong>in</strong>g s<strong>in</strong>θ and cosθ from the identitiess<strong>in</strong>θ =√1−cos2θ2and cosθ =√1+cos2θ2Example 1.13 Discuss and sketch the graph of the equation xy = 1.Solution .........Example 1.14 Identify and sketch the curve41x 2 −24xy +34y 2 −25 = 0Solution .........
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