Two Dimensional Co-ordinate Geometry

Two Dimensional Co-ordinate Geometry - Two Dimensional...

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Two Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Advanced Level Pure Mathematics Calculus II 9.1 Introduction 9.2 Change of Axes 9.3 Straight Lines 9.4 Equations of Lines Pairs 9.5 Circle 9.6 Parabola 9.7 Ellipse 9.8 Hyperbola 9.1 Introduction If 2 1 m , m be the gradients of two straight lines respectively, angle θ between them, the θ is given by Prepared by K. F. Ngai Page 1 9
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Two Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics tan θ = 2 1 2 1 m m 1 m m + - If the points 2 1 P , P and P are collinear, then P is said to divide the line segment 2 1 P P in the ratio 2 1 m : m or r where r = . PP P P 2 1 We have x = , r 1 rx x 2 1 + + y = r 1 ry y 2 1 + + If P divides 2 1 P P internally, r is positive. If P divides 2 1 P P externally, r is negative. Area of triangle is 1 y x 1 y x 1 y x 2 1 3 3 2 2 1 1 counter-clockwise Hence area of quadrilateral vertices are A: ), y , x ( i i i = 1, 2, 3, 4 arranged in counter- clockwise is + + + 1 1 4 4 4 4 3 3 3 3 2 2 2 2 1 1 y x y x y x y x y x y x y x y x 2 1 Area of n-sided polygon is + + + + - - 1 1 n n n n 1 n 1 n 3 3 2 2 2 2 1 1 y x y x y x y x . . . y x y x y x y x 2 1 Condition for collinearity of 3 points is 1 y x 1 y x 1 y x 3 3 2 2 1 1 = 0 Parametric Equation Given a pair of equation x = x(t), y = y(t), when we eliminate the variable t, we obtain equation f(x, y) = 0. Prepared by K. F. Ngai Page 2 P 1 (x 1 , y 1 ) P 2 (x 2 , y 2 ) P (x, y) r 1 :
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Two Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Example 1 F: axy 3 y x 3 3 - + = 0, a 0 By considering the intersection of line y = tx and curve F. Show curve F may represented parametrically by x = , t 1 at 3 2 + y = 2 2 t 1 at 3 + 9.2Change of Axes 1. Translation of axes Take new axes O X and O Y parallel to OX and OY where O (h, k). Let the old and new co-ordinates of P be (x, y) and (x , y ) we have x = x + h y = y + k 2. Rotation of co-ordinates axes Prepared by K. F. Ngai Page 3 y y x x 0 (h, k) × P y y x P
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Two Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics Let the old and new co- ordinates of P be (x, y) and (X, Y) x = OP cos ( θ + φ ) = OP cos θ cos φ - OP sin θ sin φ = X cos θ + Y sin θ y = OP sin ( θ + φ ) = OP sin θ cos φ - OP cos θ sin φ = X sin θ + Y cos θ we have y x = θ θ θ - θ Y X cos sin sin cos (old) (new) Y X = θ θ - θ θ y x cos sin sin cos (new) (old)  Example 2 Give E: 31 y 18 x 16 y 3 x 4 2 2 + + - + = 0 if the origin of co-ordinates system is translated to (2, - 3), find the equation of curve in new coordinates system.
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Two Dimensional Co-ordinate Geometry - Two Dimensional...

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