Karnataka 1st PUC Mathematics Straight Lines MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – I PUC
NAME OF THE CHAPTER: STRAIGHT LINES

Slope of a Line:
 If 𝜃 is the inclination of a line 𝑙, then tan 𝜃 is called the slope or gradient of the line 𝑙.
 The slope of a line whose inclination is 90° is not defined. The slope of a line is
denoted by 𝑚. Thus, 𝒎 = 𝐭𝐚𝐧 𝜽, 𝜃 ≠ 90°
 It may be observed that the slope of 𝑥-axis is zero and slope of 𝑦-axis is not defined
 Slope of a line when coordinates of any two points 𝑃(𝑥1 , 𝑦1 ) and 𝑄(𝑥2 , 𝑦2 )
𝒚 −𝒚
on the line are given is given by 𝒎 = 𝒙𝟐 −𝒙𝟏
𝟐 𝟏

 Conditions for parallelism and perpendicularity of lines in terms of their slopes
Let 𝑚1 be the slope of line 𝑙1 and 𝑚2 be the slope of the line 𝑙2 .
o If 𝒍𝟏 ∥ 𝒍𝟐 then 𝒎𝟏 = 𝒎𝟐 and
𝟏 𝟏
o if 𝒍𝟏 ⊥ 𝒍𝟐 then 𝒎𝟏 . 𝒎𝟐 = −𝟏 or 𝒎𝟏 = − 𝒎 or 𝒎𝟐 = − 𝒎
𝟐 𝟏

Angle between two lines:
The acute angle 𝛼 between two lines 𝑙1 and 𝑙2 with slopes 𝑚1 and 𝑚2 is given by
𝑚 −𝑚
1 2
tan 𝛼 = |1+𝑚 |
1𝑚 2

Various Forms of the Equation of a Line:
Horizontal and vertical lines
 Equation of line parallel to 𝑥-axis is of the form 𝑦 = 𝑘 where 𝑘 is real constant.
In particular, equation of 𝑥-axis is 𝒚 = 𝟎.
 Equation of line parallel to 𝑦-axis is of the of form 𝑥 = 𝑘 where 𝑘 is real constant.
In particular, equation of 𝑦-axis is 𝒙 = 𝟎
Point-slope form:
Equation of line passing through a given point (𝑥1 , 𝑦1 ) and having slope 𝑚 is given by
𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏 )

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Two-point form:
Equation of line passing through two given points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) is given by
𝒚 −𝒚 𝒙−𝒙 𝒚−𝒚
𝒚 − 𝒚𝟏 = (𝒙𝟐 −𝒙𝟏 ) (𝒙 − 𝒙𝟏 ) or 𝒙 −𝒙𝟏 = 𝒚 −𝒚𝟏
𝟐 𝟏 𝟐 𝟏 𝟐 𝟏

Slope-Intercept form
 Equation of line with slope 𝑚 and 𝑦-intercept 𝑐 is given by 𝒚 = 𝒎𝒙 + 𝒄
 Equation of line with slope 𝑚 and 𝑥-intercept 𝑑 is given by 𝒚 = 𝒎(𝒙 − 𝒅)
Intercept form
𝒙 𝒚
 Equation of line with 𝑥-intercept 𝑎 and 𝑦-intercept 𝑏 is given by 𝒂 + 𝒃 = 𝟏

General equation of a line:
The general equation of a line is given by 𝑨𝒙 + 𝑩𝒚 + 𝑪 = 𝟎.
Note:
 If 𝐴 = 0 then the line is parallel to 𝑥-axis.
 If 𝐵 = 0 then the line is parallel to 𝑦-axis.
 If 𝐶 = 0 then the line is passes through origin
𝑨
 Slope, 𝒎 = − 𝑩
𝑪
 𝑥-intercept, 𝒂 = − 𝑨
𝑪
 𝑦-intercept, 𝒃 = − 𝑩
 Equation of any line parallel to 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 is of the form 𝑨𝒙 + 𝑩𝒚 + 𝒌 = 𝟎
 Equation of any line perpendicular to 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 is of the form 𝑩𝒙 − 𝐀𝒚 + 𝒌 = 𝟎

Distance of a Point from a Line:
The distance of a point from a line is the length of the perpendicular drawn from
|𝑨𝒙𝟏 +𝑩𝒚𝟏 +𝑪|
the point (𝑥1 , 𝑦1 ) to the line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 is given by 𝒅 =
√𝑨𝟐 +𝑩𝟐

Distance between parallel lines:
o Considering the lines in slope-intercept form, The distance between two parallel lines
|𝒄𝟐 −𝒄𝟏 |
𝑦 = 𝑚𝑥 + 𝑐1 and 𝑦 = 𝑚𝑥 + 𝑐2 is given by
√𝟏+𝒎𝟐

o Considering the lines in their general form the distance between two parallel lines
|𝑪𝟐 −𝑪𝟏 |
𝐴𝑥 + 𝐵𝑦 + 𝐶1 = 0 and 𝐴𝑥 + 𝐵𝑦 + 𝐶2 = 0 is given by
√𝑨𝟐 +𝑩𝟐

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 The area of the triangle bounded by the straight line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0, (𝑎, 𝑏, 𝑐 ≠ 0) and
𝟏 𝑪𝟐
the coordinate axes is 𝟐 |𝑨𝑩|

 If (ℎ, 𝑘) is the foot of the perpendicular from (𝑥1 , 𝑦1 ) to the line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 then
𝒉−𝒙𝟏 𝒌−𝒚𝟏 −(𝑨𝒙𝟏 +𝑩𝒚𝟏 +𝑪)
= =
𝑨 𝑩 𝑨𝟐 +𝑩𝟐

 If (ℎ, 𝑘) is the image of (𝑥1 , 𝑦1 ) with respect to the line 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 then
𝒉 − 𝒙𝟏 𝒌 − 𝒚𝟏 −𝟐(𝑨𝒙𝟏 + 𝑩𝒚𝟏 + 𝑪)
= =
𝑨 𝑩 𝑨 𝟐 + 𝑩𝟐

OTHER FORMULAE STUDIED IN PREVIOUS CLASSES RELATED TO
COORDINATE GEOMETRY:
 Distance between the points 𝑃(𝑥1 , 𝑦1 ) and 𝑄(𝑥2 , 𝑦2 ) is 𝑷𝑸 = √(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐
 The coordinates of a point dividing the line segment joining the points
mx2  nx1 my2  ny1 
(𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) internally, in the ratio 𝑚 ∶ 𝑛 are  , .
 mn mn 

 In particular, if 𝑚 = 𝑛, the coordinates of the mid-point of the line segment joining the
 x  x2 y1  y2 
points (𝑥1 , 𝑦1 ) and (𝑥2 , 𝑦2 ) are  1 , 
 2 2 

 Area of the triangle wtih vertices (𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 ) and (𝑥3 , 𝑦3 ) is given by
1
𝑨= |𝒙𝟏 (𝒚𝟐 − 𝒚𝟑 ) + 𝒙𝟐 (𝒚𝟑 − 𝒚𝟏 ) + 𝒙𝟑 (𝒚𝟏 − 𝒚𝟐 )|
2
Remark:
If the area of the triangle ABC is zero, then three points A, B and C lie on a line, i.e.,
they are collinear.

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SLOPE
1. The slope of the line which makes 𝟒𝟓° with positive 𝒙-axis measured anticlockwise
is (Easy)
(A) −1 (B) 1 (C) 0 (D) not defined.
𝝅
2. The slope of the line which makes 𝟒 with positive 𝒚-axis measured anticlockwise

is (Easy)
(A) −1 (B) 1 (C) 0 (D) not defined.
3. The angle of inclination with respect to positive 𝒙-axis, of the line with slope −𝟏
is (Easy)
𝜋 3𝜋
(A) −1° (B) 1° (C) 4 (D) 4 .

4. The line if its slope is zero is (Easy)
(A) Either the line is y-axis or it parallel to y-axis
(B) 𝜃 is an obtuse angle
(C) Either the line is x-axis or it parallel to x-axis
(D) 𝜃 is an acute angle
5. The slope of the line through the points (𝟑, −𝟐) and (−𝟏, 𝟒) is (Easy)
2 3 3
(A) 3 (B) 2 (C) − 2 (D) not defined.

6. The slope of the line through the points (𝟑, −𝟐) and (𝟑, 𝟒) is (Easy)
(A) 0 (B) 4 (C) −1 (D) not defined.
7. Line through the points (– 𝟐, 𝟔) and (𝟒, 𝟖) is parallel to the line through the points
(𝟖, 𝟏𝟐) and (𝒙, 𝟐𝟒),then the value of 𝒙. (Easy)
(A) 0 (B) 4 (C) −1 (D) 44.
8. Line through the points (– 𝟐, 𝟔) and (𝟒, 𝟖) is perpendicular to the line through the
points (𝟖, 𝟏𝟐) and (𝒙, 𝟐𝟒),then the value of 𝒙. (Easy)
(A) 0 (B) 4 (C) −1 (D) 44.
9. The slope of the line which makes angle 300 with positive direction of y axis
measured anticlockwise (Easy)
1
(A) √3 (B) −√3 (C) 1 (D) .
√3

10. If three points (h, 0), (a, b) and (0, k) lie on a line, then (Difficult)
𝑎 𝑏 𝑎 𝑏 ℎ 𝑘 ℎ 𝑘
(A) ℎ + 𝑘 = 1 (B) ℎ − 𝑘 = 1 (C) 𝑎 + 𝑏 = 1 (D) 𝑎 − 𝑏 = 1.

11. A line cuts off equal intercepts on the co-ordinate axes. The angle made by this line
with the positive direction of 𝒙-axis is [KCET 2019] (Average)
(A) 1200 (B) 450 (C) 1350 (D) 900
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12. If the co-ordinates of the points 𝑨 and 𝑩 be (𝟏, 𝟎) and (𝟐, √𝟑), then the angle made by
the line 𝑨𝑩 with 𝒙-axis is (Average)
(A) 30° (B) 45° (C) 60° (D) 75°
13. The line passes through (𝟏, 𝟎) and (−𝟐, √𝟑) makes an angle of ...... with 𝒙-axis
(Average)
(A) 60° (B) 120𝑜 (C) 150∘ (D) 135∘
14. The line passing through the points (𝟑, −𝟒) and (-2, 6) and a line passing through
(−𝟑, 𝟔) and (𝟗, −𝟏𝟖) are (Easy)
(A) Perpendicular (B) Parallel
(C) Makes an angle 60∘ with each other (D) None of these
15. The angle between the lines 𝒍𝟏 passing through the points (𝟐, −𝟏) and (−𝟐, 𝟑) and
𝒍𝟐 passing through (𝟒, 𝟏) and (−𝟐, 𝟏) are (Difficult)
(A) 0𝑜 (B) 120𝑜 (C) 90∘ (D) 45∘
Various forms of Equation of Line
16. Equation of the line passing through (−𝟐, 𝟑) with slope −4 is (Easy)
(A) 4𝑥 + 𝑦 − 11 = 0 (B) 4𝑥 + 𝑦 + 5 = 0
(C) 𝑥 + 4𝑦 − 11 = 0 (D) 𝑥 + 4𝑦 + 5 = 0.
17. Equation of the line through the points (1,-1) and (3,5) is (Easy)
(A) 𝑦 = 3𝑥 + 4 (B) 𝑦 = 3𝑥 − 4 (C) 3𝑥 + 𝑦 − 4 = 0 (D)3𝑥 + 𝑦 + 4 =0.
𝟏
18. Equation of the lines for which 𝐭𝐚𝐧 𝜽 = 𝟐 where 𝜽 is the inclination of the line

and 𝒚-intercept is 4 is (Easy)
(A) 𝑥 − 2𝑦 + 8 = 0 (B) 𝑦 = 𝑥 + 8 (C) 𝑥 + 2𝑦 + 8 = 0 (D) −𝑥 + 2𝑦 + 4 = 0.
19. Equation of the line passing through the point (𝟐, 𝟑) with inclined with x-axis
at an angle of 𝟒𝟓° 𝑖𝑠 (Average)
(A) 𝑥 − 𝑦 + 5 = 0 (B) 𝑥 − 𝑦 − 1 = 0 (C) 𝑥 + 𝑦 + 8 = 0 (D) 𝑥 − 𝑦 + 1 = 0
20. Equation of the line intersecting the x-axis at a distance of 3 units to the left
of origin with slope –2 is (Average)
(A) 2𝑥 − 2𝑦 − 6 =0 (B) 𝑦 = 2𝑥 + 6 (C) 2𝑥 − 𝑦 + 6 = 0 (D) 2𝑥 + 𝑦 + 6 = 0 .
21. The vertices of 𝜟𝑷𝑸𝑹 are 𝑷(𝟐, 𝟏), 𝑸(– 𝟐, 𝟑) and 𝑹(𝟒, 𝟓), then equation of the median
through the vertex R. (Average)
(A) 3𝑥 + 4𝑦 − 8 = 0 (B) 3𝑥 + 4𝑦 + 8 = 0
(C) 3𝑥 − 4𝑦 + 8 = 0 (D) 3𝑥 − 4𝑦 − 8 = 0.

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22. A line meets 𝒙-axis and 𝒚-axis at the points 𝑨 and 𝑩 respectively.
If the middle point of 𝑨𝑩 be (𝒙𝟏 , 𝒚𝟏 ), then the equation of the line is (Average)
(A) 𝑦1 𝑥 + 𝑥1 𝑦 = 2𝑥1 𝑦1 (B) 𝑥1 𝑥 + 𝑦1 𝑦 = 2𝑥1 𝑦1
(C) 𝑦1 𝑥 + 𝑥1 𝑦 = 𝑥1 𝑦1 (D) 𝑥1 𝑥 + 𝑦1 𝑦 = 𝑥1 𝑦1
23. The equation of a straight line passing through ( −𝟐, 𝟑) and cutting an intercept equal
in magnitude but opposite in sign from the axes is given by (Average)
(A) 𝑥 − 𝑦 + 5 = 0 (B) 𝑥 + 𝑦 − 5 = 0 (C) 𝑥 − 𝑦 − 5 = 0 (D) 𝑥 + 𝑦 + 5 = 0
24. The points 𝑨(𝟏, 𝟑) and 𝑪(𝟓, 𝟏) are the opposite vertices of rectangle.
The equation of line passing through other two vertices and of gradient 2, is (Difficult)
(A) 2𝑥 + 𝑦 − 8 = 0 (B) 2𝑥 − 𝑦 − 4 = 0 (C) 2𝑥 − 𝑦 + 4 = 0 (D) 2𝑥 + 𝑦 + 7 = 0
25. The equation of line whose mid point is (𝒙𝟏 , 𝒚𝟏 ) in between the axes, is (Average)
𝑥 𝑦 𝑥 𝑦 1 𝑥 𝑦
(A) 𝑥 + 𝑦 = 2 (B) 𝑥 + 𝑦 = 2 (C) 𝑥 + 𝑦 = 1 (D) None of these
1 1 1 1 1 1

26. The equation of the line passing through (𝟒, −𝟔) and makes an angle 𝟒𝟓∘ with
positive 𝒙-axis, is (Average)
(A) 𝑥 − 𝑦 − 10 = 0 (B) 𝑥 − 2𝑦 − 16 = 0
(C) 𝑥 − 3𝑦 − 22 = 0 (D) None of these
27. The equation of a straight line passing through the points (-5, -6) and (3, 10), is
(Easy)
(A) 𝑥 − 2𝑦 = 4 (B) 2𝑥 − 𝑦 + 4 = 0 (C) 2𝑥 + 𝑦 = 4 (D) None of these
28. The equation of a line through the intersection of lines 𝒙 = 𝟎 and 𝒚 = 𝟎
and through the point (𝟐, 𝟐), is (Average)
(A) 𝑦 = 𝑥 − 1 (B) 𝑦 = −𝑥 (C) 𝑦 = 𝑥 (D) 𝑦 = −𝑥 + 2
29. A straight the makes an angle of 𝟏𝟑𝟓∘ with the 𝒙-axis and cuts 𝒚-axis at a distance -5
from the origin. The equation of the line is (Average)
(A) 2𝑥 + 𝑦 + 5 = 0 (B) 𝑥 + 2𝑦 + 3 = 0
(C) 𝑥 + 𝑦 + 5 = 0 (D) 𝑥 + 𝑦 + 3 = 0
30. The equation of the line which cuts off an intercept 3 units on 𝑶𝑿 and
an intercept −𝟐 unit on 𝑶𝒀, is (Average)
𝑥 𝑦 𝑥 𝑦 𝑥 𝑦 𝑥 𝑦
(A) 2 − 3 = 1 (B) 3 + 2 = 1 (C) 2 + 3 = 1 (D) 3 − 2 = 1

31. If the intercept made by the line between the axis is bisected at the point (5, 2), then
its equation is (Average)
(A) 5𝑥 + 2𝑦 = 20 (B) 2𝑥 − 5𝑦 = 20
(C) 5𝑥 − 2𝑦 = 20 (D) 2𝑥 + 5𝑦 = 20

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32. Equation of the straight line making equal intercepts on the axes and passing
through the point (𝟐, 𝟒) is (Average)
(A) 4𝑥 − 𝑦 − 4 = 0 (B) 2𝑥 + 𝑦 − 8 = 0
(C) 𝑥 + 𝑦 − 6 = 0 (D) 𝑥 + 2𝑦 − 10 = 0
General Form of Equation of Line
33. The values of 𝒌 for which the line (𝒌– 𝟑)𝒙– (𝟒– 𝒌𝟐 )𝒚 + 𝒌𝟐 – 𝟕𝒌 + 𝟔 = 𝟎 is
parallel to the 𝒚-axis (Average)
(A) 4 (B) ±2 (C) 3 (D) 6, 1.
34. The values of k for which the line (𝒌– 𝟑)𝒙– (𝟒– 𝒌𝟐 )𝒚 + 𝒌𝟐 – 𝟕𝒌 + 𝟔 = 𝟎 is
parallel to the x-axis, (Average)
(A) 4 (B) ±2 (C) 3 (D) 6, 1.
35. The values of k for which the line (𝒌– 𝟑)𝒙– (𝟒– 𝒌𝟐 )𝒚 + 𝒌𝟐 – 𝟕𝒌 + 𝟔 = 𝟎 is
passing through the origin. (Average)
(A) 4 (B) ±2 (C) 3 (D) 6, 1.
36. Perpendicular distance from the origin to line √𝟑𝒙 + 𝒚 + 𝒌 = 𝟎 is 5 units , then 𝒌 is
(Average)
(A) ±6 (B) 0 (C) ±5 (D) ±10
37. The x-intercept of the line 𝟑𝒙– 𝟒𝒚 + 𝟏𝟎 = 𝟎 is (Easy)
5 10 3
(A) 2 (B) − 3 (C) 4 (D) 2.

38. The distance of the point (– 𝟏, 𝟏) from the line 𝟏𝟐(𝒙 + 𝟔) = 𝟓(𝒚– 𝟐). (Average)
55 60
(A) 13 (B) 5 (C) 13 (D) 13.

39. The distance between the lines 𝟑𝒙 − 𝟒𝒚 + 𝟕 = 𝟎 and 𝟑𝒙 − 𝟒𝒚 + 𝟓 = 𝟎 is (Easy)
2 5 12 60
(A) 5 (B) 2 (C) 25 (D) 13

40. Equation of the line passing through (1,2) and parallel to 𝒚 = 𝟑𝒙 − 𝟏 is (Easy)
(A) 𝑦 + 2 = 𝑥 + 1 (B) 𝑦 + 2 = 3(𝑥 + 1)
(C) 𝑦 − 2 = 3(𝑥 − 1) (D) 𝑦 − 2 = 𝑥 − 1
41. The equation of the straight line passing through the point (𝟏, 𝟐) and
perpendicular to the line 𝒙 + 𝒚 + 𝟏 = 𝟎 is (Average)
(A) 𝑦 − 𝑥 + 1 = 0 (B) 𝑦 − 𝑥 − 1 = 0
(C) 𝑦 − 𝑥 + 2 = 0 (D) 𝑦 − 𝑥 − 2 = 0

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42. The two lines 𝒂𝟏 𝒙 + 𝒃𝟏 𝒚 + 𝒄𝟏 = 𝟎 and 𝒂𝟐 𝒙 + 𝒃𝟐 𝒚 + 𝒄𝟐 = 𝟎 where 𝒃𝟏 , 𝒃𝟐 ≠ 𝟎 are
𝑎1 𝑏
Statement I: Parallel if = 1 (Easy)
𝑎2 𝑏2

Statement II: Perpendicular if 𝑎1 𝑎2 − 𝑏1 𝑏2 = 0
(A) Both I and II are true (B) only I is true
(C) only II is true (D) Both I and II are false
43. A line passes through (𝟐, 𝟐) and perpendicular to the line 𝟑𝒙 + 𝒚 = 𝟑.
It’s 𝒚-intercept is [KCET 2015, 2023] (Average)
1 2 4
(A) 3 (B) 3 (C) 3 (D) 1

44. If the straight lines 𝟐𝒙 + 𝟑𝒚 − 𝟑 = 𝟎 and 𝒙 + 𝒌𝒚 + 𝟕 = 𝟎 are perpendicular, then the value
of 𝒌 is [KCET 2016] (Average)
2 3 2 3
(A) 3 (B) 2 (C) − 3 (D) − 2

45. Equation of line passing through the point (𝟏, 𝟐) and perpendicular to the
line 𝒚 = 𝟑𝒙 − 1 is [KCET 2017] (Easy)
(A) 𝑥 + 3𝑦 − 7 = 0 (B) 𝑥 + 3𝑦 + 7 = 0 (C) 𝑥 + 3𝑦 = 0 (D) 𝑥 − 3𝑦 = 0
46. The equation of the line parallel to the line 𝟑𝒙 − 𝟒𝒚 + 𝟐 = 𝟎 and
passing through (−𝟐, 𝟑) is [KCET 2018] (Easy)
(A) 3𝑥 − 4𝑦 + 18 = 0 (B) 3𝑥 − 4𝑦 − 18 = 0
(C) 3𝑥 + 4𝑦 + 18 = 0 (D) 3𝑥 + 4𝑦 − 18 = 0
47. The two lines 𝒍𝒙 + 𝒎𝒚 = 𝒏 and 𝒍′ 𝒙 + 𝒎′ 𝒚 = 𝒏′ are perpendicular if
[KCET 2020] (Average)
(A) 𝑙𝑚′ + 𝑚𝑙 ′ = 0 (B) 𝑙𝑙 ′ + 𝑚𝑚′ = 0 (C) 𝑙𝑚′ = 𝑚𝑙 ′ (D) 𝑙𝑚 + 𝑙 ′ 𝑚′ = 0
48. The equation of straight line which passes through the point (𝒂𝐜𝐨𝐬𝟑 𝜽, 𝒂𝐬𝐢𝐧𝟑 𝜽) and
perpendicular to 𝒙𝐬𝐞𝐜 𝜽 + 𝒚𝐜𝐨𝐬𝐞𝐜𝜽 = 𝒂 is [KCET 2021] (Difficult)
𝑥 𝑦
(A) 𝑎 + 𝑎 = 𝑎cos 𝜃 (B) 𝑥cos 𝜃 − 𝑦sin 𝜃 = 𝑎cos 2𝜃

(C) 𝑥cos 𝜃 + 𝑦sin 𝜃 = 𝑎cos 2𝜃 (D) 𝑥cos 𝜃 − 𝑦sin 𝜃 = −𝑎cos 2𝜃
49. If the straight line 𝟐𝒙 − 𝟑𝒚 + 𝟏𝟕 = 𝟎 is perpendicular to the line passing through the
point (𝟕, 𝟏𝟕) and (𝟏𝟓, 𝜷), then 𝜷 equals [KCET 2022] (Average)
(A) -5 (B) 5 (C) 29 (D) -29
50. The angle between the line 𝒙 + 𝒚 = 𝟑 and the line joining the points (𝟏, 𝟏) and (−𝟑, 𝟒) is
[KCET 2024] (Average)
1 1 2
(A) tan−1 (7) (B) tan−1 (− 7) (C) tan−1 (7) (D) tan−1 (7)

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51. A line passes through (−𝟏, −𝟑) and perpendicular to 𝒙 + 𝟔𝒚 = 𝟓. Its 𝒙 intercept is
[KCET 2025] (Average)
1 1
(A) (B) -2 (C) 2 (D) −
2 2

52. The line 𝐋𝟐 which passes through (𝟑, −𝟏) divides the line 𝐋𝟏 joining the two points
(−𝟏, 𝟐) and (𝟑, 𝟔) in the ratio 𝟏: 𝟑 internally, then the equation of 𝑳𝟐 is
[KCET 2026] (Difficult)
(A) 𝑥 − 3𝑦 − 6 = 0 (B) 𝑥 − 𝑦 + 3 = 0
(C) 4𝑥 + 3𝑦 − 9 = 0 (D) 4𝑥 + 3𝑦 + 9 = 0
53. 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 (Average)
(A) 𝑥 + 5𝑦 = ±5 (B) 𝑥 − 5𝑦 = 5√2 (C) 𝑥 − 5𝑦 = 5 (D) 𝑥 + 5𝑦 = ±5√2
54. The equation of the straight line passing through the point (𝟑, 𝟐) and perpendicular to
the line 𝒚 = 𝒙 is (Average)
(A) 𝑥 − 𝑦 = 5 (B) 𝑥 + 𝑦 = 5 (C) 𝑥 + 𝑦 = 1 (D) 𝑥 − 𝑦 = 1
55. If the coordinates of the points 𝑨, 𝑩, 𝑪 be (−𝟏, 𝟓), (𝟎, 𝟎) and (𝟐, 𝟐) respectively and 𝑫 be
the middle point of 𝑩𝑪, then the equation of the perpendicular drawn from 𝑩 to the
line 𝑨𝑫 is (Difficult)
(A) 𝑥 + 2𝑦 = 0 (B) 2𝑥 + 𝑦 = 0 (C) 𝑥 − 2𝑦 = 0 (D) 2𝑥 − 𝑦 = 0
56. A line passes through the point of intersection of 𝟐𝒙 + 𝒚 = 𝟓 and 𝒙 + 𝟑𝒚 + 𝟖 = 𝟎 and
parallel to the line 𝟑𝒙 + 𝟒𝒚 = 𝟕 is (Average)
(A) 3𝑥 + 4𝑦 + 3 = 0 (B) 3𝑥 + 4𝑦 = 0 (C) 4𝑥 − 3𝑦 + 3 = 0 (D) 4𝑥 − 3𝑦 = 3
57. If the equation 𝒚 = 𝒎𝒙 + 𝒄 and 𝒙𝐜𝐨𝐬 𝜶 + 𝒚𝐬𝐢𝐧 𝜶 = 𝒑 represents the same straight line, then
(Difficult)
(A) 𝑝 = 𝑐√1 + 𝑚2 (B) 𝑐 = 𝑝√1 + 𝑚2
(C) 𝑐𝑝 = √1 + 𝑚2 (D) 𝑝2 + 𝑐 2 + 𝑚2 = 1
58. Equation of line passing through (𝟏, 𝟐) and perpendicular to 𝟑𝒙 + 𝟒𝒚 + 𝟓 = 𝟎 is
(Average)
(A) 3𝑦 = 4𝑥 − 2 (B) 3𝑦 = 4𝑥 + 3 (C) 3𝑦 = 4𝑥 + 4 (D) 3𝑦 = 4𝑥 + 2
59. The equations of the lines through the origin making an angle of 𝟔𝟎𝒐 with the line
𝒙 + 𝒚√𝟑 + 𝟑√𝟑 = 𝟎 are (Average)
(A) 𝑦 = 0, 𝑥 − 𝑦√3 = 0 (B) 𝑥 = 0, 𝑥 − 𝑦√3 = 0
(C) 𝑥 = 0, 𝑥 + 𝑦√3 = 0 (D) 𝑦 = 0, 𝑥 + 𝑦√3 = 0

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60. Equation of a line passing through (𝟏, −𝟐) and perpendicular to the line
𝟑𝒙 − 𝟓𝒚 + 𝟕 = 𝟎 is (Average)
(A) 5𝑥 + 3𝑦 + 1 = 0 (B) 3𝑥 + 5𝑦 + 1 = 0
(C) 5𝑥 − 3𝑦 − 1 = 0 (D) 3𝑥 − 5𝑦 + 1 = 0
𝟑
61. If the slope of a line passing through the point 𝑨(𝟑, 𝟐) be 𝟒, then the points on the line

which are 5 units away from 𝑨, are (Difficult)
(A) (6, 6), (0, −2) (B) (7, 5), (−1, −1)
(C) (5, 7), (−1, −1) (D) (7, 7), (1, 1)
62. The acute angle between the lines 𝒙 = 𝟑 and 𝒚 = √𝟑𝒙 + 𝟗 is (Easy)
(A) 30° (B) 60° (C) 45° (D) 90°
𝒙 𝒚 𝒙 𝒚
63. Angle between the lines 𝒂 + 𝒃 = 𝟏 and 𝒂 − 𝒃 = 𝟏 is (Difficult)
𝑏 2𝑎𝑏 𝑎2 −𝑏2
(A) 2tan−1 𝑎 (B) tan−1 𝑎2 +𝑏2 (C) tan−1 𝑎2 +𝑏2 (D) None of these

64. The angle between the two lines 𝒚 − 𝟐𝒙 = 𝟗 and 𝒙 + 𝟐𝒚 = −𝟕, is (Average)
(A) 60° (B) 30° (C) 90° (D) 45°
65. If the lines 𝒚 = (𝟐 + √𝟑)𝒙 + 𝟒 and 𝒚 = 𝒌𝒙 + 𝟔 are inclined at an angle 𝟔𝟎∘ to each other,
then the value of 𝒌 will be (Average)
(A) 1 (B) 2 (C) -1 (D) -2
66. If 𝜽 is the acute angle between the lines 𝒂𝟏 𝒙 + 𝒃𝟏 𝒚 + 𝒄𝟏 = 𝟎 and 𝒂𝟐 𝒙 + 𝒃𝟐 𝒚 + 𝒄𝟐 = 𝟎, then
𝐭𝐚𝐧 𝜽 is (Average)
𝑎 𝑏 −𝑎 𝑏 𝑎 𝑎 +𝑏 𝑏 𝑎 𝑏 −𝑎 𝑏 𝑎 𝑏 +𝑎 𝑏
(A) |𝑎1 𝑎2 +𝑏2 𝑏1 | (B) |𝑎1 𝑏2−𝑎1𝑏2 | (C) |𝑎1 𝑎1 +𝑏2𝑏2 | (D) |𝑎1 𝑎1 −𝑏2𝑏2 |
1 2 1 2 1 2 2 1 1 2 1 2 1 2 1 2

67. The angle between the straight lines 𝒙 − 𝒚√𝟑 = 𝟓 and √𝟑𝒙 + 𝒚 = 𝟕 is (Average)
(A) 90° (B) 60° (C) 75° (D) 30°
𝒙 𝒚
68. The length of the perpendicular from the point ( 𝒃, 𝒂 ) to the line 𝒂 − 𝒃 = 𝟏, is

(Average)
𝑎2 −𝑎𝑏+𝑏 2 𝑏 2 −𝑎𝑏−𝑎2 𝑎2 +2𝑎𝑏−𝑏2
(A) | √𝑎2 | (B) | √𝑎2 | (C) | √𝑎2 | (D) None of these
+𝑏2 +𝑏 2 +𝑏2

69. The distance of the point of intersection of the lines 𝟐𝒙 − 𝟑𝒚 + 𝟓 = 𝟎 and 𝟑𝒙 + 𝟒𝒚 = 𝟎
from the line 𝟓𝒙 − 𝟐𝒚 = 𝟎 is (Difficult)
130 13 130
(A) 17√29 (B) 7√29 (C) (D) None of these
17

70. The vertex of an equilateral triangle is (𝟐, −𝟏) and the equation of its base in
𝒙 + 𝟐𝒚 = 𝟏. The length of its sides is (Difficult)
3 2 4 3
(A) 2√15 (B) (C) 3√3 (D)
√15 √5

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𝒙 𝒚
71. The length of the perpendicular drawn from origin upon the straight line 𝟑 − 𝟒 = 𝟏 is

(Average)
2 1 2 2
(A) 2 5 (B) 3 5 (C) 4 5 (D) 3 5

72. The distance of the point (−𝟐, 𝟑) from the line 𝒙 − 𝒚 = 𝟓 is (Easy)
(A) 5√2 (B) 2√5 (C) 3√5 (D) 5√3
73. Distance between the lines 𝟓𝒙 + 𝟑𝒚 − 𝟕 = 𝟎 and 𝟏𝟓𝒙 + 𝟗𝒚 + 𝟏𝟒 = 𝟎 is (Average)
35 1 35 35
(A) (B) 3√34 (C) 3√34 (D) 2√34
√34

74. The length of perpendicular from the point (𝒂𝐜𝐨𝐬 𝜶, 𝒂𝐬𝐢𝐧 𝜶) upon the straight line
𝒚 = 𝒙𝐭𝐚𝐧 𝜶 + 𝒄, 𝒄 > 𝟎 is (Difficult)
(A) 𝑐cos 𝛼 (B) 𝑐sin2 𝛼 (C) 𝑐sec 2 𝛼 (D) 𝑐cos 2 𝛼
75. Distance between the two parallel lines 𝒚 = 𝟐𝒙 + 𝟕 and 𝒚 = 𝟐𝒙 + 𝟓 is (Easy)
√5 2 2 1
(A) 2 (B) 5 (C) (D)
√5 √5

76. The foot of the coordinates drawn from (𝟐, 𝟒) to the line 𝒙 + 𝒚 = 𝟏 is (Average)
1 3 1 3 4 1 3 1
(A) (3 , 2) (B) (− 2 , 2) (C) (3 , 2) (D) (4 , − 2)

77. The image of a point 𝑨(𝟑, 𝟖) in the line 𝒙 + 𝟑𝒚 − 𝟕 = 𝟎, is (Difficult)
(A) (-1,-4) (B) (-3, -8) (C) (1,-4) (D) (3,8)
78. If (−𝟐, 𝟔) is the image of the point (𝟒, 𝟐) with respect to line 𝑳 = 𝟎, then 𝑳 = (Difficult)
(A) 3𝑥 − 2𝑦 + 5 (B) 3𝑥 − 2𝑦 + 10 (C) 2𝑥 + 3𝑦 − 5 (D) 6𝑥 − 4𝑦 − 7
79. The triangle formed by the lines 𝒙 + 𝒚 − 𝟒 = 𝟎, 𝟑𝒙 + 𝒚 = 𝟒, 𝒙 + 𝟑𝒚 = 𝟒 is (Difficult)
(A) Isosceles (B) Equilateral (C) Right-angled (D) None of these
80. The area of the triangle formed by the line 𝒙𝐬𝐢𝐧 𝜶 + 𝒚𝐜𝐨𝐬 𝜶 = 𝐬𝐢𝐧 𝟐𝜶 and the
coordinates axes is (Difficult)
(A) sin 2𝛼 (B) cos 2𝛼 (C) 2sin 2𝛼 (D) 2cos 2𝛼

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STRAIGHT LINES
1 2 3 4 5 6 7 8 9 10
B A D C C D D B D A
11 12 13 14 15 16 17 18 19 20
C C B B D B B A D D
21 22 23 24 25 26 27 28 29 30
C A A B A A B C C D
31 32 33 34 35 36 37 38 39 40
D C B C D D B B A C
41 42 43 44 45 46 47 48 49 50
B B C C A A B C B C
51 52 53 54 55 56 57 58 59 60
D C D B C A B D B A
61 62 63 64 65 66 67 68 69 70
B A A C C A D B A A
71 72 73 74 75 76 77 78 79 80
A A C A C B A A A A

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