Search & Filtering
The normal to the parabola y2= 4ax at the point P(at2 , 2at) meets the x-axis at A. Find the equation of the locus of the midpoint of AP as t varies.
a'. Vector (b' vector ×c' vector)=1/(a vector ×b vector)c vector
. In each part, sketch the graph of the equation in 3-space. (a) x = y 2 (b) z = x 2 (c) y = z 2
Reduce each equation to standard form. Then find the coordinates of the center, the foci, the ends of the major and minor axes, and the ends of each latus rectum. Sketch the curve
(a) x2 + 4y2 + 6x + 16y + 21 = 0
(b) 16x2 + 4y2 + 32x − 16y − 32 = 0
(c) 4x2 + 8y2 − 4x − 24y − 13 = 0
The normal at a point P of an ellipse, of which S is a focus, meets the ellipse again in Q, and the normal at Q meets the major axis in L. A line through L parallel to the line P Q meets SP in R. Prove that |SQ| = |SR|.
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.
