Analytic Geometry Answers

Draw the ellipse with the equation: 2𝑥 2 + 𝑦 2 + 4𝑥 − 9𝑦 = 7

Find the equation of the ellipse whose foci are at(0,-5) and (0,5) and the length of its major axis is 14

find the area of a triangle with vertices E(1,3) F(-4,-3) G(5,-4)

Show that the closed sphere with centre (7,3,2)and radius 10 in R³ is contained in the open cube P = {(x, y,z ):|x − 2 |<11,|y− 3|<11, |z − 7 |<11}.

Find the cylindrical coordinates of the points where the Cartesian coordinates are

i.(6,6,8)

ii.(√2,1,1)

Express the following surfaces in spherical coordinates.

i.xz=3

ii.x²+y²-z²=1

Prove that

(i) the three points A(1, 4, 2), B(3, 2, 4) and C(5, 0, 6) are collinear,

(ii) the four points P(2, ;1, 1), Q(1, 3, ;2), R((2, 1, ;3) and S(3, 2, 0) are coplanar

Prove that

(i) the three points A(1, 4, 2), B(3, 2, 4) and C(5, 0, 6) are collinear,

(ii) the four points P(2, ;1, 1), Q(1, 3, ;2), R((2, 1, ;3) and S(3, 2, 0) are coplanar.

7. If a = i + 3j j k, b = 2i + 4j j 2k and c = =i + 2j + 4k, find a number λ such that

d = a + b + λc is parallel to the yz-plane.

If a and b denote the vectors OA and OB, indicate on the same diagram the vectors OC and OD denoted by a+b and a-b. Draw on another diagram the vector OE denoted by a+2b