Analytic Geometry Answers

Calculate the area of the triangle through the points (5,3,2),(-2,7,-1) and (4,-2,6)

Find the equation of the plane through the points (1,-2,4) ,(3,1,-2) and (2,-1,6)

A line passes through the points (7,-4,8) and (9,0,5)

i) Determine the parametric and symmetric equations of the line

ii)Find any other point on the line

iii)Show that the line does not pass through the origin

Find the area of the triangle determined by the points (2,-1,0),(2,0,4) and (0,2,3)

Find the perpendicular distance from point (1,-1,4) to the plane 2x-3y-2z=7

Find the equation of the plane through the points (1,1,1),(-4,0,2) and (3,-1,0)

Find the sum of two vectors if one has a direction of 60 degree and a magnitude of 5 and the other has a direction of 240 and a magnitude of 2?

16x²+4y²+32x-16y-32=0

center,foci, ends of major and minor axis, ends of latus rectum

1.The points P(ap2 , 2ap) and Q(aq2 , 2aq) lie on the parabola y 2 = 4ax. Prove that if P Q is a focal chord then the tangents to the curve at P and Q intersect at right angles at a point on the directrix.

2. The tangents at the points P(ap2 , 2ap) and Q(aq2 , 2aq) on the parabola y 2 = 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.

3. The coordinates of the ends of a focal chord of the parabola y 2 = 4ax are (x1, y1) and (x2, y2). Show that x1x2 = a 2 and y1y2 = −4a 2 .

4. Prove that the line x − 2y + 4a = 0 touches the parabola y 2 = 4ax, and find the coordinates of P, the point of contact. If the line x − 2y + 2a = 0 meets the parabola in Q, R, and M is the mid-point of QR, prove that PM is parallel to the axis of x, and that this axis and the line through M perpendicular to it meet on the normal at P to the parabola