Introduction To Three Dimensional Geometry

More Topic from Mathematics

Question 1

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is

  • 3x2/2

  • x3/8

  • x2/2

  • πx2

Solution

C.

x2/2


area space equals space 1 half space straight x squared space sin space straight theta
straight A subscript max space equals space 1 half space straight x squared space open parentheses at space sin space straight theta space equals 1 comma space straight theta space equals straight pi over 2 close parentheses
Question 2

If (a, a2 ) falls inside the angle made by the lines y =x/2, x >0 and y = 3x, x > 0, then a belongs to

  • (0,1/2)

  • (3, ∞)

  • (1/2, 3)

  • (-3, -1/2)

Solution

C.

(1/2, 3)

Question 3

If PS is the median of the triangle with vertices P(2,1), Q(6,-1) and R (7,3), then equation of the line passing through (1,-1) and parallel to PS is

  • 4x-7y - 11 =0

  • 2x+9y+7=0

  • 4x+7y+3 = 0

  • 2x-9y-11  =0

Solution

B.

2x+9y+7=0

Coordinate of S  = open parentheses fraction numerator 7 plus 6 over denominator 2 end fraction comma fraction numerator 3 minus 1 over denominator 2 end fraction close parentheses space equals space open parentheses 13 over 2 comma 1 close parentheses
[∵ S is mid-point of line QR]
A slope of the line PS is -2/9.
Required equation passes through (1,-1) and parallel to PS is
straight y space plus space 1 space equals space fraction numerator negative 2 over denominator 9 end fraction left parenthesis straight x minus 1 right parenthesis
rightwards double arrow space 2 straight x space plus space 9 straight y space plus 7 space equals space 0

Question 4

Let a,b,c and d be non-zero numbers. If the point of intersection of the lines 4ax +2ay +c= 0 and 5bx +2by +d = 0 lies in the fourth quadrant and is equidistant from the two axes, then

  • 2bc-3ad =0 

  • 2bc+3ad =0

  • 2ad-3bc =0

  • 3bc+2ad=0

Solution

C.

2ad-3bc =0

Let coordinate of the intersection point in the fourth quadrant be (α, -α) lies on both lines 4ax +2ay +c =0 and 5bx +2by +d =0
therefore space 4 aα space minus 2 aα space plus straight c space equals 0 space rightwards double arrow straight alpha space equals space fraction numerator negative straight c over denominator 2 straight a end fraction space.... space left parenthesis straight i right parenthesis
5 bα space minus 2 bα space plus straight d space equals 0 space rightwards double arrow space straight alpha space equals space fraction numerator negative straight d over denominator 3 straight b end fraction space..... space left parenthesis ii right parenthesis
From space Eqs. space left parenthesis straight i right parenthesis space and space left parenthesis ii right parenthesis comma space we space get
fraction numerator negative straight c over denominator 2 straight a end fraction space equals space fraction numerator negative straight d over denominator 3 straight b end fraction
rightwards double arrow space 3 bc space equals space 2 ad
rightwards double arrow 2 ad minus 3 bc space equals space 0

Question 5

The angle between the lines whose direction cosines satisfy the equations l +m+n=0 and l2 = m2+n2 is

  • π/3

  • π/4

  • π/6

  • π/2

Solution

A.

π/3

We know that angle between two lines is 
cos space straight theta space equals space fraction numerator straight a subscript 1 straight a subscript 2 space plus straight b subscript 1 straight b subscript 2 space plus straight c subscript 1 straight c subscript 2 over denominator square root of straight a subscript 1 superscript 2 plus straight b subscript 1 superscript 2 plus straight c subscript 1 superscript 2 end root square root of straight a subscript 2 superscript 2 plus straight b subscript 2 superscript 2 plus straight c subscript 2 superscript 2 end root end fraction
l +m +n= 0
⇒ l = - (m+n)
⇒ (m+n)2 = l2
⇒ m2 +n2 +2mn = m2 +n2
[∵ l2 = m2 +n2, given]
⇒ 2mn = 0
when m = 0 ⇒ l =-n
Hence, (l, m, n) is (1,0-1)
When n =0, then l =-m
Hence, (l,m,n) is (1,0-1)
Cos space straight theta space equals space fraction numerator 1 plus 0 plus 0 over denominator square root of 2 space end root space straight x square root of 2 end fraction space equals space 1 half rightwards double arrow straight theta space equals space straight pi over 3