Co-ordinate Geometry5.

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• Slope of a line in terms of inclination 2

is, m = tan

• Slope of X-axis is 0.• Slope of Y-axis is not defined.• If l1 and l2 are two lines with slopes m1 and m2 respectively, then

m1 = m2 when l1 l2m1 × m2 = –1, when l1 l2

• Slope of a line passing through A(x1, y1) and B(x2, y2) is

2 1

2 1

––

y yx x = 1 2

1 2

––

y yx x or [x1 x2]

Equation of lines :• Equation of X-axis is y = 0. The slope of x-axis is 0.• Equation of a line parallel to X-axis is y = k. (where k is constant)• Equation of Y-axis is x = 0. The slope y-axis is not defined.• Equation of a line parallel to Y-axis is x = k (where k is constant).• Equation of a line with slope-m and y-intercept c is y = mx + c. (Slope-intercept form)• Equation of a line with slope m passing through origin is y = mx.• Equation of a line passing through the point P(x1, y1) having slope m is y – y1 = m(x – x1).

[Slope-point form]• Equation of a line passing through the points A(x1, y1) and B(x2, y2) is

1

1 2

––

x xx x = 1

1 2

––

y yy y [Two-point form] [x1 x2]

• Equation of a line with x-intercept a and y-intercept b is +x ya b = 1 [Double-intercept

form] [a 0, b 0]• The general equation of a straight line is Ax + By + C = 0, A, B, C R

• Slope of the line Ax + By + C = 0 is– Coefficient ofCoefficient of

xy

=– AB

• To find the x-intercept of the line Ax + By + C = 0, put y = 0 Ax + c = 0 x =–Ac

• To find the y-intecept of the line Ax + By + C = 0, put x = 0 By + c = 0 y =–Bc

MULTIPLE CHOICE QUESTIONS (MCQ's)(Each question carries one mark)

1. Slope of line AB, where A = (–2, 1) and B = (0, 3) is ........ .

a –1 b 0 c 1 d 12

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2. The vertices of ABC are A(3, –4), B(5, 7) and C(–4, 5). Then the slope of seg BC is ........ .

a 29

b 92

c – 29

d – 92

3. The slope and y-intercept of the line y = 3x – 5 are ........ .a (–5, –3) b (–5, 3) c (–3, –5) d (3, –5)

4. The equation of line 2y – 3x + 5 = 0, in slope intercept form is ........ .

a y =– 3 5–2 2

x b y =– 3 5+2 2

x c y =3 5–2 2

x d y =3 5+2 2

x

5. The point (4, –3) lies on the line 5x + 8y = c, then the value of c is ........ .

a 4 b – 4 c 14

d –14

6. The slope of a line is 3 and y-intercept is –4; then the equation of line is ........ .

a y = 3x + 4 b y = 3x – 4 c y = –3x + 4 d y = –3x – 4

7. The slope of line y + 3 =1 ( – 5)2

x is ........ .

a 1–2

b –2 c 12

d 2

8. Slope of the line 3(x + 3) = y – 1, ........ .

a 3 b 13

c –3 d –13

9. The equation of a line passing through (–3, 7) with slope m =12

is ........ .

a x + 2y + 17 = 0 b x + 2y – 17 = 0 c x – 2y + 17 = 0 d x – 2y – 17 = 0

10. The equation of a line passing through origin with slope m = 32

is ........ .

a 3x – 2y = 0 b 3x + 2y = 0 c 2x + 3y = 0 d 2x – 3y = 0

11. The equation of a line passing through (2, 3) and (4, 7) is ........ .

a 2x + y – 1 = 0 b 2x – y – 1 = 0 c 2x + y + 1 = 0 d 2x – y + 1 = 0

12. The x and y-intercepts of the line –3 4x y

= 1 are respectively ........ .

a (3, –4) b (3, 4) c (–3, 4) d (–3, –4)13. The x and y intercepts of the line y – x = –5 are respectively ........ .

a (–5, –5) b (–5, 5) c (5, 5) d (5, –5)14. The slope of the line y – 3x – 6 = 0 is ........ .

a 13

b 1–3

c 3 d –3

15. The equation of line 4x – y – 7 = 0 in the double intercept form is ........ .

a

+ = 17 ( – 7)4

x yb

+ = 17 74

x y c

+ = 1– 7 (– 7)4

x yd

+ = 1– 7 74

x y

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16. Equation of X-axis is ........ .a x = 0 b y = 0 c x = k d y = k

17. Equation of a line parallel to X-axis and passing through (–3, 4) is ........ .a x = – 3 b x = 3 c y = – 4 d y = 4

18. Equation of a line parallel Y-axis and passing through (–3, 5) is ........ .a x = 3 b x = – 3 c y = 5 d y = – 5

19. The equation of a line passing through the point (2, 7) and whose y-intercept is 3 is ........ .a 2x – y + 3 = 0 b 2x – y – 3 = 0 c 2x + y + 3 = 0 d 2x + y – 3 = 0

20. The equation of a line parallel to X-axis and at a distance of 'a' unit above it is .........a x = a b y = a c x = 0 d y = 0

21. For Y-axis and every line parallel to Y-axis the inclination will be .........a 0º b 90º c 30º d 45º

22. Equation of line passing through point A and point B can be formed by .........a slope-point form b two-point formc double-intercept form d slope-intercept form

23. Slopes of parallel lines are .........a zero b equal c 1 d 2

24. Slope of a line is equal to .........a b tan c cos d sin

Answers

1. (c) m =3 – 1

0 – (– 2) =22

= 1

2. (a) m(BC) = 5 – 7– 4 – 5

=– 2– 9

=29

3. (d) y = 3x – 5, comparing with y = mx + cm = 3, c = –5

4. (c) 2y = 3x – 5

y =3 5–2 2

x

5. (b) 5(4) + 8(–3) = c c = 20 – 24 = – 4

6. (b) y = mx + c y = 3x + (– 4) y = 3x – 4

7. (c) y + 3 =1 ( – 5)2

x ,

comparing with y – y1 = m(x – x1)

m =12

8. (a) 3(x + 3) = y – 1, which can be writtenas y – 1 = 3(x + 3),comparing with y – y1 = m(x – x1)

m = 3

9. (c) y – 7 =1 ( + 3)2

x

2y – 14 = x + 3 x – 2y + 17 = 0

10. (a) The equation of a line passing throughorigin with slope m is y = mx

y =32

x

2y = 3x 3x – 2y = 0

11. (b) Equation of line passing through(2, 3) and (4, 7) is,

(x1, y1) (x2, y2)

1

1 2

––

x xx x = 1

1 2

––

y yy y

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– 22 – 4x

=– 3

3 – 7y

– 2– 2

x =

– 3– 4

y

– 21

x =

– 32

y

2x – 4 = y – 3 2x – y – 4 + 3 = 0 2x – y – 1 = 0

12. (a) + = 13 (– 4)x y , comparing with + = 1x y

a ba = 3 (x-intercept), b = –4 (y-intercept)

13. (d) y – x = –5 5 = x – y

– = 15 5x y

+ = 15 (–5)x y

comparing with + = 1x ya b

a = 5, b = 5

14. (c) y – 3x – 6 = 0 y = 3x + 6, comparing with y = mx + c, m = 3

15. (a) 4x – y = 7

4 – = 17 7x y

+ = 17 (– 7)4

x y

16. (b) On the X-axis, every point will havey-co-ordinate

y = 0 is the equation of X-axis.17. (d) Equation of a line parallel to X-axis

is y = kSince the line passes through (–3, 4)the equation of line is y = 4

18. (b) Equation of a line parallel to Y-axisis x = k.Since the line passes through (–3, 5)the equation of line is x = –3.

19. (a) The equation of a line passing through(2, 7) and (0, 3) is

1

1 2

––

x xx x = 1

1 2

––

y yy y

1

1 2

––

x xx x = 1

1 2

––

y yy y

=– 2

2 – 0x

=– 7

7 – 3y

=– 21

x =

– 72

y

2x – 4 = y – 7 2x – y – 4 + 7 = 0 2x – y + 3 = 0

20. (b) y = a21. (b) = 90º22. (b) two-point form23. (b) equal24. (b) tan

Problems For Practice

1. Slope of line PQ, when P = (0, –5) and Q = (1, 2) is ........ .

a –7 b 0 c 7 d 17

2. Slope of Y-axis is ........ .a –1 b 0 c 1 d Not-defined

3. The vertices of PQR are P (3, 7), Q (5, 11) and R (–2, 8), then slope of seg QR is ........ .

a 73

b 37

c – 73

d – 37

4. The slope and y-intercept of the line y = –2x + 3 are respectively ........ .a (–2, –3) b (–2, 3) c (–3, –2) d (–3, 2)

5. The equation of line x + 2y – 4 = 0 in slope-intercept form is ........ .

a 1= – 22

y x b –1= – 22

y x c 1= + 22

y x d –1= + 22

y x

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6. The point (–2, –3) is a point on the line 2y = kx + 5, then the value of k is ........ .

a – 211

b –112

c 112

d 211

7. The slope of the line y – 5 = 2(x – 7) is ........ .

a 12

b 2 c –12

d –2

8. The line PQ intersects Y-axis at (0, 3) with slope –12 then the equation of line PQ is ........ .

a 1= + 32

y x b 1= – 32

y x c –1= + 32

y x d –1= – 32

y x

9. The equation of a line passing through (–2,–3) with slope m =35

is ........ .

a 3x – 5y + 9 = 0 b 3x + 5y – 9 = 0 c 3x – 5y – 9 = 0 d 3x + 5y + 9 = 010. The equation of a line passing through (0, 5) and (5, 6) is ........ .

a x – 5y + 25 = 0 b x + 5y – 25 = 0 c x – 5y – 25 = 0 d x + 5y + 25 = 0

11. The x and y-intercepts of the line +– 4 –1x y

= 1 are respectively ........ .

a (–4, 1) b (–4, –1) c (4, 1) d (4, –1)

12. The x and y-intercepts of line y =2 – 43

x are ........ respectively .

a (6, 4) b (6, – 4) c (–6, 4) d (– 6, – 4)13. Slope of the line 2x – 3y = 7 is ........ .

a – 32

b 3

2c – 2

3d 2

314. The equation of the line 3x + y = 4 in the double-intercept form is ........ .

a

+ = 14 43

x y b

+ = 14 (– 4)3

x yc

+ = 13 44

x yd

+ = 13 (– 4)4

x y

15. Equation of a line parallel to Y-axis is ........ .a x = 0 b x = k c y = 0 d y = k

Answers1 - c 2 - d 3 - b 4 - b 5 - d 6 - c7 - b 8 - c 9 - c 10 - a 11 - b 12 - b13 - d 14 - a 15 - b