Mathematics_Today_-_October_2016

SOME IMPORTANT RESULTS
z Area of the triangle formed by the lines y = m 1 x + c 1 ,

y = m 2 x + c 2 and y = m 3 x + c 3 , is^1

2

122

12

∑ −

−

()cc

mm

.

z Area of the triangle made by the line ax + by + c = 0

with the co-ordinate axes is c

ab

2

2| |

.

z Area of the rhombus formed by the lines

ax ± by ± c = 0 is^2

c^2

ab

.

z Area of the parallelogram formed by the lines
a 1 x + b 1 y + c 1 = 0; a 2 x + b 2 y + c 2 = 0, a 1 x + b 1 y + d 1 = 0
and a 2 x + b 2 y + d 2 = 0 is ()( )dcd c
ab a b

112 2

12 21

−−

−

.

z The foot of the perpendicular (h, k) from
(x 1 , y 1 ) to the line ax + by + c = 0 is given
by hx
a

ky

b

ax by c

ab

− = − =− ++

+

1111

22

(). Hence,

the coordinates of the foot of perpendicular is

bx aby ac

ab

ay abx bc

ab

(^211)
22
(^211)
22
−−

• −−


• 
  

  ⎛
  ⎝
  ⎜
  ⎞
  ⎠
  , ⎟.
  z Area of parallelogram A
  =pp^12
  sinθ, where p^1 and p^2
  are the distances between parallel sides and θ is the
  angle between two adjacent sides.
  z The equation of a line whose mid-point is (x 1 , y 1 )
  in between the axes is x
  x
  y
  11 y
  += 2.
  z The equation of a straight line which makes a triangle
  with the axes of centroid (x 1 , y 1 ) is x
  x
  y
  33 y
  1
  11
  +=.
  PROBLEMS
  Single Correct Answer Type

  
  

1. A line L is perpendicular to the line 5x – y = 1
   and the area of the triangle formed by the line L and
   coordinate axes is 5. The equation of the line L is

(a) x + 5y = 5 (b) xy+=±^552
(c) x – 5y = 5 (d) xy−^552 =

2. If the coordinates of the points A, B, C and D be
   (a, b), (a′, b′), (–a, b) and (a′, –b′) respectively, then the

equation of the line bisecting the line segments AB and

CD is

(a) 22 ay′ − bx ab ab= − ′′

(b)^22 ay− b x′ =ab a b− ′′

(c) 22 ay− b x′ =a b ab′ − ′

(d) none of these

3. If the middle points of the sides BC, CA and AB of
   the triangle ABC be (1, 3), (5, 7) and (–5, 7) respectively,
   then the equation of the side AB is
   (a) x – y – 2 = 0 (b) x – y + 12 = 0
   (c) x + y – 12 = 0 (d) none of these


4. The equation of the line perpendicular to the line
   x
   a

y

b

− = 1 and passing through the point at which it

cuts x-axis, is

(a) x

a

y

b

a

b

++= 0 (b)

x

b

y

a

b

a

+=

(c) x

b

y

a

+= 0 (d) x

b

y

a

a

b

+=

5. The equation of the line bisecting the line segment
   joining the points (a, b) and (a′, b′) at right angle, is
   (a) 22 ()()aax bby a b a b− ′ + − ′ =+22 2 2− ′ − ′
   (b) ()()aax bby a b a b− ′ + − ′ =+22 2 2− ′ − ′
   (c) 22 ()()aax bby a b a b− ′ + − ′ = ′^2222 + ′ −−
   (d) none of these


6. The equation of the lines which passes through
   the point (3, –2) and are inclined at 60° to the line
   31 xy+=are
   (a) yxy+=20 3, −−−^2330 =
   (b) xxy−20 3= , − ++^2330 =
   (c) 32330 xy−−− = (d) none of these


7. Equation of the line passing through (–1, 1) and
   perpendicular to the line 2x + 3y + 4 = 0 is
   (a) 2(y – 1) = 3(x + 1) (b) 3(y – 1) = –2(x + 1)
   (c) y – 1 = 2(x + 1) (d) 3(y – 1) = x + 1


8. The intercept cut off from y-axis is twice that from
   x-axis by the line and line is passes through (1, 2), then
   its equation is
   (a) 2x + y = 4 (b) 2x + y + 4 = 0
   (c) 2x – y = 4 (d) 2x – y + 4 = 0


9. The equation of line whose mid point is (x 1 , y 1 ) in
   between the axes, is
   (a) x
   x

y

11 y

+= 2 (b) x

x

y

11 y

1

2

+=