Chapter 1 Topics in Analytic Geometry

MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 7yP(x,y))••F ′ (−c,0)•F(c,0)xLet P(x,y) be any point on the ellipse. Since the sum of the distances from P to the fociis 2a, it follows thatPF ′ +PF = 2aso√(x+c)2 +y 2 + √ (x−c) 2 +y 2 = 2aTransposing the second radical to the right side of the equation and squaring yields(x+c) 2 +y 2 = 4a 2 −4a √ (x−c) 2 +y 2 +(x−c) 2 +y 2and, on simplifying,√(x−c)2 +y 2 = a− c a xSquaring both sides again yieldswhich, by (1.5), can be written asx 2a 2 + y2a 2 −c 2 = 1x 2a + y22 b = 1 (1.7)2Conversely it can be shown that any point whose coordinates satisfying (1.7) has 2a asthe sum of its distances from the foci, so that such a point is on the ellipse.Example 1.3 Sketch the graph of the ellipseSolution .........4x 2 +18y 2 = 36.Example 1.4 Find an equation of the ellipse with foci (0,±2) and major axis with endpoints(0,±4).Solution .........