Ace the Fundas of Mathematics with Anshul Singhal Sir

Assignment
By Anshul Singhal Sir

Practice Questions

Question Slope of a Line & Different forms of

based on Y
Equation of Straight Line
B

Slope (Gradient) of a line c

X'

The trigonometrical tangent of the angle that a line X
A O
makes with the positive direction of the x-axis in
anticlockwise sense is called the slope or gradient of the Y'
line. The slope of a line is generally denoted by m. Thus,
m = tan  . The equation of a line with slope m and the x-
intercept d is y  m(x  d ) .
(4) Intercept form: If a straight line cuts x-axis at A and
Y Y the y-axis at B then OA and OB are known as the
intercepts of the line on x-axis Y
B
B and y-axis respectively.

B
X AYO
Then, equation of a straight line
X X OY A X cutting off intercepts a and b on b
x–axis and y–axis respectively is A
X' X
x y O a
(1) Slope of line parallel to x – axis is m  tan 0  0 .o  1.
a b
(2) Slope of line parallel to y – axis is m  tan 90 o   . Y'
(3) Slope of the line equally inclined with the axes is 1 or If given line is parallel to X
– 1. axis, then X-intercept is undefined.
(4) Slope of the line through the points A(x1, y1 ) and If given line is parallel to Y axis, then Y-intercept is
y 2  y1 undefined.
B(x 2 , y 2 ) is taken in the same order.
x 2  x1 (5) Two point form: Equation of the line through the
a y 2  y1
(5) Slope of the line ax  by  c  0, b  0 is  . points A (x1, y1 ) and B(x 2 , y2 ) is, (y  y1 )  (x  x1 ) .
b x 2  x1
(6) Slope of two parallel lines is equal.
In the determinant form it is gives as Y L
(7) If m1 and m 2 be the slopes of two perpendicular
lines, then m1.m2  1 . x y 1 B
(x2, y2)
x1 y1 1 = 0

(8) m can be defined as tan  for 0     and   . x2 y2 1
2
is the equation of line.
Equations of straight line in different A O X

forms
(x1,y1)

(1) Slope form: Equation of a line through the origin and (6) Normal or perpendicular form : The equation of
having slope m is y = mx. the straight line upon which the length of the
(2) One point form or Point slope form: Equation of a perpendicular from the origin is p and this perpendicular
line through the point (x1, y1 ) and having slope m makes an angle  with x-axis is x cos   y sin   p .
Y
is y  y1  m (x  x1 ) .
(3) Slope intercept form: Equation of a line (non- B
vertical) with slope m and cutting off an intercept c on p P
the y-axis is y  mx  c . X'

X
O A

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(7) Symmetrical or parametric or distance form of (C) 3y – 5x + 15 = 0 (D) None of these
the line : Equation of a line passing through (x1, y1 ) and
making an angle  with the positive direction of x-axis is Q.7 If the line y = mx + c passes through the
x  x1 y  y1 points (2, 4) and (3, –5), then -
  r , where r is the distance between the
cos sin (A) m = –9, c = –22 (B) m = 9, c = 22
point P (x, y) and A(x1, y1 ) .
(C) m = –9, c = 22 (D) m = 9, c = –22
Y
(x1,y1) r Q.8 The equation of the line inclined at an angle
A  P(x, y)
of 60º with x-axis and cutting y-axis at the
point (0, –2) is -

X' X
O (A) 3y=x–2 3 (B) y = 3 x – 2

Y' (C) 3y=x+2 3 (D) y = 3 x + 2

The co-ordinates of any point on this line may be
taken as (x 1  r cos , y1  r sin ) , known as parametric
Q.9 The equation of a line passing through the
co-ordinates. ‘r’ is called the parameter.
origin and the point (a cos, a sin ) is-
(A) y = x sin  (B) y = x tan  
 (C) y = x cos   (D) y = x cot 
Q.1 The angle made by the line joining the points
(1, 0) and (–2, 3 ) with x axis is -
Q.10 Slope of a line which cuts intercepts of equal
(A) 120º (B) 60º
lengths on the axes is -
(C) 150º (D) 135º
(A) –1 (B) 2

Q.2 If A(2,3), B(3,1) and C(5,3) are three points, (C) 0 (D) 3
then the slope of the line passing through
A and bisecting BC is - Q.11 The intercept made by line x cos + y sin = a
(A) 1/2 (B) –2 on y axis is -
(C) –1/2 (D) 2 (A) a (B) a cosec
(C) a sec (D) a sin
Q.3 If the vertices of a triangle have integral
coordinates, then the triangle is - Q.12 The equation of the straight line which passes
(A) Isosceles (B) Never equilateral
through the point (1, –2) and cuts off equal
(C) Equilateral (D) None of these
intercepts from axes will be-
(A) x + y =1 (B) x – y = 1
Q.4 The equation of a line passing through the
(C) x + y + 1 = 0 (D) x – y – 2 = 0
point (–3, 2) and parallel to x-axis is -
(A) x – 3 = 0 (B) x + 3 = 0
Q.13 The intercept made by a line on y-axis is double
(C) y – 2 = 0 (D) y + 2 = 0
to the intercept made by it on x-axis. If it passes
Q.5 If the slope of a line is 2 and it cuts an intercept through (1, 2) then its equation-
– 4 on y-axis, then its equation will be - (A) 2x + y = 4 (B) 2x + y + 4 = 0
(A) y – 2x = 4 (B) x = 2y – 4 (C) 2x – y = 4 (D) 2x – y + 4 = 0
(C) y = 2x – 4 (D) None of these
Q.14 If the point (5, 2) bisects the intercept of a line
Q.6 The equation of the line cutting of an intercept between the axes, then its equation is-
–3 from the y-axis and inclined at an angle (A) 5x + 2y = 20 (B) 2x + 5y = 20
tan–1 3/5 to the x axis is - (C) 5x – 2y = 20 (D) 2x – 5y = 20
(A) 5y – 3x + 15 = 0 (B) 5y – 3x = 15 

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Q.15 If the point (3,–4) divides the line between the Q.21 For a variable line x/a + y/b = 1, a + b = 10, the
x-axis and y-axis in the ratio 2 : 3 then the locus of mid point of the intercept of this line
equation of the line will be - between coordinate axes is -
(A) 2x + y = 10 (B) 2x – y = 10 (A) 10x + 5y = 1 (B) x + y = 10
(C) x + 2y = 10 (D) x – 2y = 10 (C) x + y = 5 (D) 5x + 10 y = 1

Q.16 The equation to a line passing through the Q.22 If a line passes through the point P(1,2) makes
point (2, –3) and sum of whose intercept on an angle of 45º with the x-axis and meets the
the axes is equal to –2 is - line x + 2y – 7 = 0 in Q, then PQ equals -
(A) x + y + 2 = 0 or 3x + 3y = 7
2 2 3 2
(B) x + y + 1 = 0 or 3x – 2y = 12 (A) (B)
3 2
(C) x + y + 3 = 0 or 3x – 3y = 5
(D) x – y + 2 = 0 or 3x + 2y = 12 (C) 3 (D) 2

Q.17 The line bx + ay = 3ab cuts the coordinate axes Q.23 A line passes through the point (1, 2) and makes
at A and B, then centroid of OAB is- 60º angle with x axis. A point on this line at a
distance 3 from the point (1, 2) is -
(A) (b, a) (B) (a, b)
(C) (a/3, b/3) (D) (3a, 3b) (A) (–5/2, 2 – 3 3 /2)
(B) (3/2, 2+ 3 3 /2)
Q.18 The area of the triangle formed by the lines (C) (5/2, 2 + 3 3 /2)
x = 0, y = 0 and x/a + y/b = 1 is- (D) None of these
(A) ab (B) ab/2
(C) 2ab (D) ab/3 Q.24 If the points (1, 3) and (5, 1) are two opposite
vertices of a rectangle and the other two vertices
lie on the line y = 2x + c, then the value of c is -
Q.19 The equations of the lines on which the (A) 4 (B) – 4
perpendiculars from the origin make 30º angle (C) 2 (D) None of these
with x-axis and which form a triangle of area
50 Question
with axes, are - based on Angle between two Straight Lines
3
If  is the angle between the lines y  m1 x  c1 and
(A) x ± 3 y – 10 = 0
m1  m 2
y  m2 x  c2 , intersecting at A. Then,   tan 1 .
(B) 3 x + y –10 = 0 1  m1m 2

(C) x + 3 y ± 10 = 0 If  is angle between two lines, then    is also the angle

between them.
(D) None of these
(1) Angle between two straight lines when their
equations are given: The angle  between the lines
Q.20 If a perpendicular drawn from the origin on any a1 x  b1y  c1  0 and a2 x  b2y  c2  0 is given by
line makes an angle 60º with x axis. If the a2 b1  a1b 2
tan   .
line makes a triangle with axes whose area is a1a2  b1b 2

54 3 square units, then its equation is - (2) Conditions for two lines to be coincident, parallel,
perpendicular and intersecting: Two lines
(A) x + 3 y = 18 (B) 3 x + y + 18 = 0 a1 x  b1y  c1  0 and a2 x  b2y  c2  0 are,

(C) 3 x + y = 18 (D) None of these a1 b1 c1

(a) Coincident, if  
a2 b2 c2
a1 b1 c1
(b) Parallel, if  
a2 b2 c2

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a1 b1 (C) 1 (D) –1
(c) Intersecting, if 
a2 b2
(d) Perpendicular, if a1a2  b1b2  0 Q.34 The equation of line passing through (2, 3) and
perpendicular to the line adjoining the points
(–5, 6) and (–6, 5) is -
Q.25 The angle between the lines y – x + 5 = 0
(A) x + y + 5 = 0 (B) x – y + 5 = 0
and 3 x – y + 7 = 0 is - (C) x – y – 5 = 0 (D) x + y – 5 = 0
(A) 15º (B) 60º
(C) 45º (D) 75º Q.35 The equation of perpendicular bisector of the
line segment joining the points (1, 2) and
Q.26 The angle between the lines 2x + 3y = 5 and (–2, 0) is -
3x – 2y = 7 is - (A) 5x + 2y =1 (B) 4x + 6y = 1
(A) 45º (B) 30º (C) 6x + 4y =1 (D) None of these
(C) 60º (D) 90º
Q.36 If the foot of the perpendicular from the origin
Q.27 The angle between the lines 2x – y + 5 = 0 and to a straight line is at the point (3, –4). Then the
3x + y + 4 = 0 is- equation of the line is -
(A) 30º (B) 90º (A) 3x – 4y = 25 (B) 3x – 4y + 25 = 0
(C) 45º (D) 60º (C) 4x + 3y –25 = 0 (D) 4x – 3y + 25 = 0
Question
based on Equation of Parallel and Perpendicular lines
Q.28 The obtuse angle between the line y = – 2 and
y = x + 2 is -
(A) 120º (B) 135º (1) Equation of a line which is parallel to ax  by  c  0
(C) 150º (D) 160º is ax  by    0 .
(2) Equation of a line which is perpendicular to
ax  by  c  0 is bx  ay    0 .
Q.29 The acute angle between the lines y = 3 and
The value of  in both cases is obtained with the
y = 3 x + 9 is -
help of additional information given in the problem.
(A) 30º (B) 60º (C) 45º (D) 90º (3) If the equation of line be a sin  b cos  c , then line
(i) Parallel to it, x sin   y cos   d

   
Q.30 Orthocenter of the triangle whose sides are
 
given by 4x – 7y + 10 = 0, x + y – 5 = 0 & (ii)Perpendicular to it is x sin   y cos   d .
7x + 4y – 15 = 0 is - 2 2
(A) (–1, –2) (B) (1, –2)
(C) (–1, 2) (D) (1, 2) Q.37 Equation of the line passing through the point
(1, –1) and perpendicular to the line 2x – 3y = 5
is -
Q.31 The angle between the lines x – 3y + 5 = 0
(A) 3x + 2y – 1 = 0
and y-axis is -
(B) 2x + 3y + 1 = 0
(A) 90º (B) 60º (C) 30º (D) 45º
(C) 3x + 2y – 3 = 0
(D) 3x + 2y + 5 = 0
Q.32 If the lines mx + 2y + 1 = 0 and 2x + 3y + 5 = 0
are perpendicular then the value of m is -
Q.38 The equation of the line passing through the
(A) –3 (B) 3 (C) –1/3 (D) 1/3
point (c, d) and parallel to the line ax + by + c = 0
is -
Q.33 If the line passing through the points (4, 3) and
(A) a(x + c) + b(y + d) = 0
(2, ) is perpendicular to the line y = 2x + 3,
(B) a(x + c) – b(y + d) = 0
then  is equal to -
(C) a(x – c) + b(y – d) = 0
(A) 4 (B) –4
(D) None of these
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Q.47 The equation of the line which passes through
Q.39 The equation of a line passing through the (a cos3, a sin3) and perpendicular to the line
point (a, b) and perpendicular to the line x sec + ycosec = a is -
ax + by + c = 0 is - (A) x cos + y sin = 2a cos2
(A) bx – ay + (a2 – b2) = 0 (B) x sin – y cos = 2a sin2
(B) bx – ay – (a2 – b2) = 0 (C) x sin + y cos = 2a cos2
(C) bx – ay = 0 (D) xcos – y sin = a cos2

(D) None of these Equation of straight lines through
Question
based on
(x1, y1) making an angle  with
Q.40 The line passes through (1, –2) and perpendicular y = mx + c
to y-axis is - 
(A) x + 1 = 0 (B) x – 1 = 0
The equation of the straight lines which pass through a
(C) y – 2 = 0 (D) y + 2 = 0 given point (x1, y1 ) and make a given angle  with given
m  tan 
Q.41 The equation of a line passing through (a, b) straight line y  mx  c are y  y1  (x  x1 ) .
1  m tan 
and parallel to the line x/a + y/b = 1 is -
(A) x/a + y/b = 0 (B) x/a + y/b = 2
(C) x/a + y/b = 3 (D) x/a + y/b + 2 = 0 Q.48 The equation of the lines which passes through
the point (3,–2) and are inclined at 60º to the
Q.42 A line is perpendicular to 3x + y = 3 and passes line 3 x + y = 1.
through a point (2, 2). Its y intercept is -
(A) y + 2 = 0, 3x–y–2–3 3=0
(A) 2/3 (B) 1/3
(C) 1 (D) 4/3 (B) 3x–y–2–3 3 =0
(C) x – 2 = 0, 3x–y+2+3 3 =0
Q.43 The equation of a line parallel to 2x – 3y = 4
which makes with the axes a triangle of area (D) None of these
12 units, is -
Q.49 (1, 2) is vertex of a square whose one diagonal
(A) 3x + 2y = 12 (B) 2x – 3y = 12
is along the x – axis. The equations of sides
(C) 2x – 3y = 6 (D) 3x + 2y = 6
passing through the given vertex are -
(A) 2x – y = 0, x + 2y + 5 = 0
Q.44 The equation of a line parallel to x + 2y = 1 and
passing through the point of intersection (B) x – 2y + 3 = 0, 2x + y – 4 = 0
of the lines x – y = 4 and 3x + y = 7 is - (C) x – y + 1 = 0, x + y – 3 = 0
(A) x + 2y = 5 (B) 4x + 8y – 1 = 0 (D) None of these
(C) 4x + 8y + 1 = 0 (D) None of these Q.50 The equation of the lines which pass through the
origin and are inclined at an angle tan–1 m to the
Q.45 The straight line L is perpendicular to the line
line y = mx + c, are-
5x – y = 1. The area of the triangle formed by
the line L and coordinate axes is 5. Then the (A) y = 0, 2mx + (1 – m2 )y = 0
equation of the line will be - (B) y = 0, 2mx + (m2 –1)y = 0
(A) x + 5y = 5 2 or x + 5y = – 5 2 (C) x = 0, 2mx + (m2 –1)y = 0
(D) None of these
(B) x – 5y = 5 2 or x – 5y = 5 2
(C) x + 4y = 5 2 or x– 2y = 5 2 Length of Perpendicular, foot of the
Question
(D) 2x + 5y = 5 2 or x + 5y = 5 2 based on
perpendicular & image of the point
with respect to line
Q.46 If (0, 0), (–2, 1) and (5, 2) are the vertices of a
triangle, Then equation of line passing through
(1) Distance of a point from a line: The length p of the
its centroid and parallel to the line x – 2y = 6 is-
perpendicular from the point (x1 , y1 ) to the line
(A) x – 2y = 1 (B) x + 2y + 1 = 0
| ax 1  by 1  c |
(C) x – 2y = 0 (D) x – 2y + 1 = 0 ax  by  c  0 is given by p  .
a2  b 2

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 Length of perpendicular from origin to the line Q.53 The length of perpendicular from the origin on
c the line x/a + y/b = 1 is -
ax  by  c  0 is .
a b
2 2
b a
(A) (B)
 Length of perpendicular from the point (x1 , y1 ) a b
2 2
a  b2
2

to the line x cos   y sin   p is| x 1 cos  y1 sin  p | . ab

(C) (D) None of these
(2) Distance between two parallel lines: Let the two
a  b2
2
parallel lines be ax  by  c1  0 and ax  by  c2  0 .
First Method: The distance between the lines is Q.54 The distance between the lines 5x + 12y + 13 = 0
| c1  c2 |
d . ax + by + c1 = 0 and 5x + 12y = 9 is -
(a 2  b 2 )
(A) 11/13 (B) 22/17
d (C) 22/13 (D) 13/22

Q.55 The distance between the parallel lines

ax + by + c2 = 0
y = 2x + 4 and 6x = 3y + 5 is -

Second Method: The distance between the lines is (A) 17/ 3 (B) 1

d , ax + by + c1 = 0 (C) 3/ 5 (D) 17 5 /15
(a 2  b 2 )
Q.56 The foot of the perpendicular drawn from the
ax + by + c2 = 0 point (7, 8) to the line 2x + 3y – 4 = 0 is -
O (0, 0)  23 2   23 
(A)  ,  (B) 13, 
 13 13   13 
where (i)  | c1  c2 | , if they be on the same side of
 23 2   2 23 
origin. (C)   ,  (D)   , 
(ii)  | c1 |  | c2 | , if the origin O lies between  13 13   13 13 
them.
Third method: Find the coordinates of any point on Q.57 The coordinates of the point Q symmetric to
one of the given line, preferably putting x  0 or y  0 . the point P (–5, 13) with respect to the line
Then the perpendicular distance of ax + by + c1 = 0 2x – 3y – 3 = 0 are -
this point from the other line is the (A) (11, –11) (B) (5, –13)
required distance between the lines. .O (0, 0)
(C) (7, –9) (D) (6, –3)
Distance between two parallel
lines ax  by  c1  0 , ax + by + c2 = 0
c2
Question Lines passing through the Point of
based on
c1  Intersection of two lines
k
kax  kby  c2  0 is .
a2  b 2
If equation of two lines P  a1 x  b1y  c1  0 and
Distance between two non-parallel lines is always zero.
Q  a2 x  b2 y  c2  0 , then the equation of the lines
Q.51 The length of the perpendicular from the origin passing through the point of intersection of these lines is
P   Q  0 or a1 x  b1 y  c1  (a 2 x  b 2 y  c 2 )  0 .
on the line 3 x – y + 2 = 0 is - Value of  is obtained with the help of the additional
(A) 3 (B) 1 information given in the problem.
(C) 2 (D) 2.5
Q.58 The line passing through the point of
Q.52 The length of perpendicular from (2, 1) on line
intersection of lines x + y – 2 = 0 and
3x – 4y + 8 = 0 is-
(A) 5 (B) 4 (C) 3 (D) 2 2x – y + 1 = 0 and origin is -
(A) 5x – y = 0 (B) 5x + y = 0
(C) x + 5y = 0 (D) x – 5y = 0

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intersection of lines y = x + 7 and x + 2y + 1 = 0,
Q.59 The equation of the line through the point of is -
intersection of the line y = 3 and x + y = 0 and (A) 2x + 5y = 0
parallel to the line 2x – y = 4 is - (B) 2x + 5y = 20
(A) 2x – y + 9 = 0 (C) 2x + 5y = 10
(B) 2x – y – 9 = 0 (D) None of these
(C) 2x – y + 1 = 0
Q.62 The equation of straight line passing through the
(D) None of these point of intersection of the lines x – y + 1 = 0
and 3x + y – 5 = 0 and perpendicular to one of
Q.60 The equation of the line passing through the them is -
point of intersection of the line 4x – 3y – 1 = 0 (A) x + y –3 = 0 or x – 3y + 5 = 0
and 5x – 2y – 3 = 0 and parallel to the line (B) x – y + 3 = 0 or x + 3y + 5 = 0
2x – 3y + 2 = 0 is - (C) x – y – 3 = 0 or x + 3y – 5 = 0
(A) x – 3y = 1 (B) 3x – 2y = 1 (D) x + y + 3 = 0 or x + 3y + 5 = 0
(C) 2x – 3y + 1 = 0 (D) 2x – y = 1

Q.61 The equation of a line perpendicular to the line

5x – 2y + 7 = 0 and passing through the point of

ANSWER KEY
Qus. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Ans. C C B C C A C B B A B C A B B B B B B A
Qus. 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
Ans. C A C B A D C B B D B B A D C A A C C D
Qus. 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Ans. B D B B A D D A C B B D C C D A A A A C
Qus. 61 62
Ans. A A

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