Archive for the ‘Co ordinate Geometry’ Category

Problem 716

Write the expression in a + bi form  (\sqrt[5]{4}(cos18 +isin18))^{5}

 

Solution:-

 

(\sqrt[5]{4}(cos18 +isin18))^{5}

=(\sqrt[5]{4})^{5}(cos(5*18)+i sin(5*18))

=4(cos 90+ isin90)

= 4(0+i(1))

=4i

 

Problem 712

A fire truck is en route to an address that is six blocks east and seven blocks south of the fire station. Using the fire station as the pole and the east direction as the polar axis, express the fire truck’s destination in polar coordinates. Round the coordinates to the nearest unit. Express θ in degrees.

 

Solution:-

 

(9, -49°)

Problem 682

Write down an (in)equality which describes the

solid ball of radius 2 centered at (1, -4, -9). It should have a

form like x^{2}+y^{2}+(z-2)^{2} -4 >= 0, where you use one of the

following symbols \leq ,< ,= ,\geq ,>.

The first blank is for the algebraic expression; the drop-down

list gives the (in)equatilty.

………………….? 0.

 

Solution:-

 

(x-1)^{2}+(y+4)^{2}+(z+9)^{2} - 4

 

<=

Problem 681

Find the center and radius of the sphere

x^{2}-6x+y^{2}-10y+z^{2}-20z = -70

Center: ( , , )

Radius:……

 

Solution:-

 

Center: (3,5,10)

 

Radius:- 8

Problem 680

Find an equation of the largest sphere with center (10, 3 , 5)

that is contained completely in the first octant.

……………………..= 0

 

Solution:-

 

\ (x-10)^{2}+(y-3)^{2}+(z-5)^{2}-9=0

Problem 679

Find an equation of the sphere that passes through the origin and whose center is (7, -3, -6).

……………………….= 0

 

Solution:-

 

x^{2}+y^{2}+z^{2}+(-14x+6y+12z)

 

Problem 678

Find the equation of the sphere if one of its diameters has

endpoints (4, -3, 1) and (6, 1, 7).

………………………= 0.

 

Solution:-

 

\ (x-5)^{2}+(y+1)^{2}+(z-4)^{2} -14 =0

Problem 677

Find the equation of the sphere centered at (-2;10;9) with radius 10.

………………………….= 0.

 

Give an equation which describes the intersection of this sphere with the plane z = 10.

 

…………………………= 0.

 

Solution:-

Find the equation of the sphere centered at (-2;10;9) with radius 10.

(x+2)^{2}+(y-10)^{2}+(z-9)^{2}-100= 0.

 

Give an equation which describes the intersection of this sphere with the plane z = 10.

 

(x+2)^{2}+(y-10)^{2}-99 = 0.

 

 

 

Problem 676

What do the following equations represent in R3?

Match the two sets of letters:

a. a vertical plane

b. a horizontal plane

c. a plane which is neither vertical nor horizontal

A. 7x+8y = -3

B. x = -4

C. y = 0

D. z = 10

 

Solution:-

7x+8y = -3

a vertical plane

 

x = -4

a vertical plane

y = 0

a vertical plane

 

z = 10

a horizontal plane

Problem 674

 

Determine whether the three points P = (-1, -7, 3), Q = (2, -1,12), R = (5, 5, 21) are colinear by computing the distances between

pairs of points.

Distance from P to Q:……………..

Distance from Q to R:……………

Distance from P to R:……………

Are the three points colinear (y/n)?………..

 

Solution:-

 

Distance from P to Q.

3\sqrt14

Distance from Q to R.

3\sqrt{14}

Distance from P to R.

6\sqrt{14}

Are the three points colinear (y/n).

y