Analytic Geometry Answers

The tangents at the points P(ap² , 2ap) and Q(aq² , 2aq) on the parabola y²= 4ax intersect at the point R. Given that the tangent at P is perpendicular to the chord OQ, where O is the origin, find the equation of the locus of R as p varies.

Reduce the following quadratic form to standard form and find its principal axes:

4x^2-4xy+y^2

5: P is a point on the parabola whose ordinate

equals its abscissa. A normal is drawn to the parabola at

P to meet it again at Q. If S is the focus of the parabola

then the product of the slopes of SP and SQ is-

Write down the parametric coordinates of any point on each of the following parabolas.

(a) y² = 8x

b) y² = 24x

c) y² = -16x

d) x² = 6y

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