Analytic Geometry Answers

Find the co-ordinate of the centre of the circle and it radius if it equation is

1. x^2 +y^2 -10x-12y+6=0

2. -2X^2-2y^2-18y +9=0

Find the section of the conicoid (x² /2) − (y ²/ 3 )= 2z by the plane x− 2y+ z = 1. What conic does this section represent? Justify your answer.

Find the area of the triangle with the given vertices A(1, 3), B(-3, 5), and C with C = 2A. (3) (3.2) Use (3.1) to find the coordinates of the point D such that the quadrilateral ABCD is a paral- (3) lelogram.

a.) Find the area of the triangle with the given vertices A(1,3), B(-3,5), and C with C = 2A.

(b) Use (a) to find the coordinates of the point D such that the quadrilateral ABCD is a parallelogram?

The planes pie1 and pie2 have cartesian equations x-y+z+1=0 and 2x+y-z-1=0 respectively.

(a) Show that these two planes are not parallel.

(b) Find a parametric equation of their line of intersection

(c)Give a cartesian equation of plane pie3 passing through point A(1,-1,-1) and perpendicular to pie1 and pie2.

Find the equation of the right circular cone whose vertex is (1,−1,2), the axis is

(x−1)/2 =( y+1)/1 = (z−2)/-2

and the semi-vertical angle is 45◦ .

Find the vertices, eccentricity, foci and asymptotes of the hyperbola (x²/8) − (y² /4) = 1. Also trace it. Under what conditions on λ the line x+λy = 2 will be tangent to this hyperbola? Explain geometrically.

Find the distance of the point of intersection of the line

(x-2)/1 = (y+3)/-1 = z/3

and the plane 2x- 3y +4z+ 4=0 from the origin.

Calculate the work done by a Force of 2𝑖 − 𝑗 − 3𝑘 in moving an object from (2, 1, 3) to (9, 4, 6) where displacement is in meters

Represent the forces 𝐹₁ = 3𝑗 − 𝑖 and 𝐹₂ = 3𝑖 + 5𝑗, on graph paper, showing both forces acting on an object at the point (0, 0). Complete the diagram to show the magnitude and direction of the resultant force 𝐹𝑅.