Analytic Geometry Past Papers PDF

Problems of Section # 1 - Analytic Geometry —- November 4, 2024 1 Section one Example 1.1 Show the relation between the Cartesian and po- lar coordinate of the following curves: (x2 + y 2 )2 = x2 − y 2 Example 1.2 Show t...

Problems of Section # 1 - Analytic Geometry —- November 4, 2024 1 Section one Example 1.1 Show the relation between the Cartesian and po- lar coordinate of the following curves: (x2 + y 2 )2 = x2 − y 2 Example 1.2 Show the relation between the Cartesian and po- lar coordinate of the following curves: r2 (cos2 (θ) − 2 cos (θ) sin (θ) + 4 sin2 θ) = 1 Example 1.3 Show the relation between the Cartesian and po- lar coordinate of the following curves: 9x2 + 4y 2 = 36 Example 1.4 Show the relation between the Cartesian and po- lar coordinate of the following curves: 2 r= 4 + 5 cos (θ) Example 1.5 What is the geometrical locus of a point P (x, y) move with 3 units from the origin. Example 1.6 A point P move on the straight line 3x+y−1 = 0 such that the point Q divide the distance between OP from inside by 2 : 1, where O is the origin, find the geometrical locus of Q. 1 Problems of Section # 2 - Analytic Geometry —- November 4, 2024 1 Section two Example 1.1 Prove that the area of triangle, its three vertices are (1, 1), (m2, m), (n2, n) equal numerically 1 (1 − m)(m − n)(1 − n) 2 Example 1.2 Evaluate the area of triangle ABC, its vertices are A(3, 2)&B(−1, 4)&C(0, 3), then find the length of perpendicular line from C to AB and the angle ∠AB̂C. Example 1.3 Prove that the following equation rep- resent a pair of straight lines 6y 2 − xy − x2 + 30y + 36 = 0 Find the angle between them and its point of intersec- tion. 1 Problems of Section # 3 - Analytic Geometry —- November 5, 2024 1 Section three Example 1.1 Find the new origin Ó(α, β) such that translate its axis to disappear the first degree from the next equation: x2 + y 2 − 4x − 6y + 4 = 0 Example 1.2 Find the new origin Ó(α, β) such that translate its axis to disappear the first degree from the next equation: 3x2 − 2xy + 4y 2 − 3x − 10y − 7 = 0 Example 1.3 Find the new origin Ó(α, β) such that translate its axis to disappear the first degree from the next equation: x2 + 4xy + 8y + 11 = 0 1 Problems of Section # 4 - Analytic Geometry —- November 5, 2024 1 Section four Example 1.1 Find the angle θ such that the axis rotate to dis- appear of the term xy from the equation: x2 − 2xy + y 2 = 4 Example 1.2 Find the angle θ such that the axes rotate to dis- appear of the term xy from the equation: x2 − 2xy + y 2 = 4 Example 1.3 If the origin moved to the point (−1, 2), then the axis rotate by angle π4 , find the new equation of the curve: 4x2 + y 2 + 8x − 4y + 7 = 0 π Example 1.4 If the axes rotate by angle 2 without change of the origin, then find the new equations of 1. The line x = 2y 2. The curve x2 − 4xy + y 2 = 1 1