Analytic Geometry Answers

If p is a parameter of positive numbers, show that all members of the family of ellipses x²/ (a² + p) + y² / (b² + p) = 1 have the same foci.

Find the Cartesian equation of the curve C traced out by a point whose coordinates, in terms of a parameter θ, are (a cos θ, b sin θ). Obtain the equations of the tangents at θ = θ₁ and θ = θ₁ + π/2. Find the coordinates of the points of intersection of the two tangents, and deduce the Cartesian equation of its locus.

Find the equations of the normal to the ellipse x² + 4y² = 65 at the points (1, 4) and (7, 2). Find the equation of the straight line joining the intersection of the normals to the origin. 

The perpendicular OY is drawn from the centre O of the ellipse x²/a² + y²/b² = 1 to any tangent. Prove that the locus of Y is (x² + y² )² = a²x² + b² y².

 Find the equations of the tangents to the ellipse x² + 4y² = 9 which are parallel to the line 2x + 3y = 0. 

Show that the tangents from the point with position vector −2i − 3j to the ellipse 4x² + 9y² = 36 are perpendicular.

Find the eccentricity of the ellipse 3x² + 4y² = 12 and the equation of the tangent to the ellipse at the point (1, 3/2). If this tangent meets the y-axis at the point G, and S and S′ are the foci of the ellipse, find the area of triangle SS′G.

The tangent at the point P(θ) on the ellipse x²/4+y²/3 = 1 passes through the point A(2, 1). Show that √3cosθ + sin θ = √3. Find all the solutions of this equation which are in the range 0 ≤ θ < 2π. Hence find the coordinates of the points of contact of the tangents to the ellipse from A. 

A point moves in the Oxy−plane so that the sum of its distances from the points S(2, 2) and S′(6, 2) is 16. Find the equation of the curve traced by the point.