Mathematics · Textbook solutions

Straight Line

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 183 questions

5.1.1 Equation of Locus

3 q

Solved Examples

Worked · 3
  1. 5.1.1 SolvedEx.1
    Find the equation of the X-axis.
  2. 5.1.1 SolvedEx.2
    Let L={P∣OP=4}L = \{P \mid OP = 4\}. Find the equation of LL.
  3. 5.1.1 SolvedEx.3
    Find the equation of the locus of points which are equidistant from A(−3,0)A(-3, 0) and B(3,0)B(3, 0). Identify the locus.

5.1.2 Shift of Origin

4 q

Solved Examples

Worked · 4
  1. If the origin is shifted to the point O′(3,2)O'(3, 2) the directions of the axes remaining the same, find the new co-ordinates of the points
    5.1.2 SolvedEx.1(a)
    A(4,6)A(4, 6)
  2. 5.1.2 SolvedEx.1(b)
    B(2,−5)B(2, -5)
  3. 5.1.2 SolvedEx.2
    The origin is shifted to the point (−2,1)(-2, 1), the axes being parallel to the original axes. If the new co-ordinates of point AA are (7,−4)(7, -4), find the old co-ordinates of point AA.
  4. 5.1.2 SolvedEx.3
    Obtain the new equation of the locus x2−xy−2y2−x+4y+2=0x^2 - xy - 2y^2 - x + 4y + 2 = 0 when the origin is shifted to (2,3)(2, 3), the directions of the axes remaining the same.

Exercise 5.1

15 q
  1. Ex 5.1 Q1
    If A(1,3)A(1, 3) and B(2,1)B(2, 1) are points, find the equation of the locus of point PP such that PA=PBPA = PB.
  2. Ex 5.1 Q2
    A(−5,2)A(-5, 2) and B(4,1)B(4, 1). Find the equation of the locus of point PP, which is equidistant from AA and BB.
  3. Ex 5.1 Q3
    If A(2,0)A(2, 0) and B(0,3)B(0, 3) are two points, find the equation of the locus of point PP such that AP=2BPAP = 2BP.
  4. Ex 5.1 Q4
    If A(4,1)A(4, 1) and B(5,4)B(5, 4), find the equation of the locus of point PP if PA2=3PB2PA^2 = 3PB^2.
  5. Ex 5.1 Q5
    A(2,4)A(2, 4) and B(5,8)B(5, 8), find the equation of the locus of point PP such that PA2−PB2=13PA^2 - PB^2 = 13.
  6. Ex 5.1 Q6
    A(1,6)A(1, 6) and B(3,5)B(3, 5), find the equation of the locus of point PP such that segment ABAB subtends right angle at PP. (∠APB=90∘)(\angle APB = 90^\circ)
  7. If the origin is shifted to the point O′(2,3)O'(2, 3), the axes remaining parallel to the original axes, find the new co-ordinates of the points
    Ex 5.1 Q7(a)
    A(1,3)A(1, 3)
  8. Ex 5.1 Q7(b)
    B(2,5)B(2, 5)
  9. If the origin is shifted to the point O′(1,3)O'(1, 3) the axes remaining parallel to the original axes, find the old co-ordinates of the points
    Ex 5.1 Q8(a)
    C(5,4)C(5, 4)
  10. Ex 5.1 Q8(b)
    D(3,3)D(3, 3)
  11. Ex 5.1 Q9
    If the co-ordinates A(5,14)A(5, 14) change to B(8,3)B(8, 3) by shift of origin, find the co-ordinates of the point where the origin is shifted.
  12. Obtain the new equations of the following loci if the origin is shifted to the point O′(2,2)O'(2, 2), the direction of axes remaining the same :
    Ex 5.1 Q10(a)
    3x−y+2=03x - y + 2 = 0
  13. Ex 5.1 Q10(b)
    x2+y2−3x=7x^2 + y^2 - 3x = 7
  14. Ex 5.1 Q10(c)
    xy−2x−2y+4=0xy - 2x - 2y + 4 = 0
  15. Ex 5.1 Q10(d)
    y2−4x−4y+12=0y^2 - 4x - 4y + 12 = 0

5.2.2 Slope of a Line

3 q

Solved Examples

Worked · 3
  1. 5.2.2 SolvedEx.1
    Find the slope of the line whose inclination is 60∘60^\circ.
  2. 5.2.2 SolvedEx.2
    Find the slope of the line which passes through the points A(2,4)A(2, 4) and B(5,7)B(5, 7).
  3. 5.2.2 SolvedEx.3
    Find the slope of the line which passes through the origin and the point A(−4,4)A(-4, 4).

5.2.3 Perpendicular Lines

2 q

Solved Examples

Worked · 2
  1. 5.2.3 SolvedEx.1
    Show that line AB is perpendicular to line BC, where A(1,2)A(1, 2), B(2,4)B(2, 4) and C(0,5)C(0, 5).
  2. 5.2.3 SolvedEx.2
    A(1,2)A(1, 2), B(2,3)B(2, 3) and C(−2,5)C(-2, 5) are vertices of △ABC\triangle ABC. Find the slope of the altitude drawn from A.

5.2.4 Angle Between Intersecting Lines

2 q

Solved Examples

Worked · 2
  1. 5.2.4 SolvedEx.1
    Find the acute angle between lines having slopes 3 and −2-2.
  2. 5.2.4 SolvedEx.2
    If the angle between two lines is 45∘45^\circ and the slope of one of the lines is 12\frac{1}{2}, find the slope of the other line.

Exercise 5.2

14 q
  1. Find the slope of each of the following lines which passes through the points :
    Ex 5.2 Q1(i)
    (a) A(2,−1)A(2, -1), B(4,3)B(4, 3)
  2. Ex 5.2 Q1(ii)
    (b) C(−2,3)C(-2, 3), D(5,7)D(5, 7)
  3. Ex 5.2 Q1(iii)
    (c) E(2,3)E(2, 3), F(2,−1)F(2, -1)
  4. Ex 5.2 Q1(iv)
    (d) G(7,1)G(7, 1), H(−3,1)H(-3, 1)
  5. Ex 5.2 Q2
    If the X and Y-intercepts of line L are 2 and 3 respectively then find the slope of line L.
  6. Ex 5.2 Q3
    Find the slope of the line whose inclination is 30∘30^\circ.
  7. Ex 5.2 Q4
    Find the slope of the line whose inclination is π4\frac{\pi}{4}.
  8. Ex 5.2 Q5
    A line makes intercepts 3 and 3 on the co-ordinate axes. Find the inclination of the line.
  9. Ex 5.2 Q6
    Without using Pythagoras theorem show that points A(4,4)A(4, 4), B(3,5)B(3, 5) and C(−1,−1)C(-1, -1) are the vertices of a right angled triangle.
  10. Ex 5.2 Q7
    Find the slope of the line which makes angle of 45∘45^\circ with the positive direction of the Y-axis measured anticlockwise.
  11. Ex 5.2 Q8
    Find the value of kk for which points P(k,−1)P(k, -1), Q(2,1)Q(2, 1) and R(4,5)R(4, 5) are collinear.
  12. Ex 5.2 Q9
    Find the acute angle between the X-axis and the line joining points A(3,−1)A(3, -1) and B(4,−2)B(4, -2).
  13. Ex 5.2 Q10
    A line passes through points A(x1,y1)A(x_1, y_1) and B(h,k)B(h, k). If the slope of the line is mm then show that k−y1=m(h−x1)k - y_1 = m(h - x_1).
  14. Ex 5.2 Q11
    If points A(h,0)A(h, 0), B(0,k)B(0, k) and C(a,b)C(a, b) lie on a line then show that ah+bk=1\frac{a}{h} + \frac{b}{k} = 1.

5.3.1 Point-slope Form

1 q

Solved Example

Worked · 1
  1. 5.3.1 SolvedEx.1
    Find the equation of the line passing through the point A(2, 1) and having slope −3-3.

5.3.2 Slope-Intercept Form

1 q

Solved Example

Worked · 1
  1. 5.3.2 SolvedEx.1
    Obtain the equation of line having slope 3 and which makes intercept 4 on the Y-axis.

5.3.3 Two-points Form

1 q

Solved Example

Worked · 1
  1. 5.3.3 SolvedEx.1
    Obtain the equation of the line passing through points A(2, 1) and B(1, 2).

5.3.4 Double-Intercept Form

1 q

Solved Example

Worked · 1
  1. 5.3.4 SolvedEx.1
    Obtain the equation of the line which makes intercepts 3 and 4 on the co-ordinate axes.

5.3 Equation of a Line in Standard Forms

9 q

Solved Examples

Worked · 9
  1. 5.3 SolvedEx.1
    The perpendicular drawn from the origin to a line has length 5 and the perpendicular makes angle with the positive direction of the X-axis. Find the equation of the line.
  2. 5.3 SolvedEx.2
    Reduce the equation 3x−y−2=0\sqrt{3}x - y - 2 = 0 into normal form. Find the values of pp and α\alpha.
  3. Find the equation of the line :
    5.3 SolvedEx.3(i)
    parallel to the XX-axis and 3 unit below it,
  4. 5.3 SolvedEx.3(ii)
    passing through the origin and having inclination 30∘30^\circ
  5. 5.3 SolvedEx.3(iii)
    passing through the point A(5,2)A(5,2) and having slope 6
  6. 5.3 SolvedEx.3(iv)
    passing through the points A(2−1)A(2-1) and B(5,1)B(5,1)
  7. 5.3 SolvedEx.3(v)
    having slope −34-\dfrac{3}{4} and yy-intercept 5,
  8. 5.3 SolvedEx.3(vi)
    making intercepts 3 and 6 on the co-ordinate axes.
  9. 5.3 SolvedEx.3(vii)
    passing through the point N(−2,3)N(-2,3) and the segment of the line intercepted between the co-ordinate axes is bisected at N.

Exercise 5.3

29 q
  1. Write the equation of the line :
    Ex 5.3 Q1(a)
    parallel to the X–axis and at a distance of 5 unit from it and above it.
  2. Ex 5.3 Q1(b)
    parallel to the Y– axis and at a distance of 5 unit from it and to the left of it.
  3. Ex 5.3 Q1(c)
    parallel to the X– axis and at a distance of 4 unit from the point (−2, 3)(-2,\ 3).
  4. Obtain the equation of the line :
    Ex 5.3 Q2(a)
    parallel to the X–axis and making an intercept of 3 unit on the Y–axis.
  5. Ex 5.3 Q2(b)
    parallel to the Y–axis and making an intercept of 4 unit on the X–axis.
  6. Obtain the equation of the line containing the point :
    Ex 5.3 Q3(a)
    A(2,−3)A(2,-3) and parallel to the Y–axis.
  7. Ex 5.3 Q3(b)
    B(4,−3)B(4,-3) and parallel to the X–axis.
  8. Find the equation of the line
    Ex 5.3 Q4(a)
    passing through the points A(2,0)A(2,0) and B(3,4)B(3,4).
  9. Ex 5.3 Q4(b)
    passing through the points P(2,1)P(2,1) and Q(2,−1)Q(2,-1)
  10. Find the equation of the line
    Ex 5.3 Q5(a)
    containing the origin and having inclination 60∘60^\circ.
  11. Ex 5.3 Q5(b)
    passing through the origin and parallel to AB, where A is (2,4)(2,4) and B is (1,7)(1,7).
  12. Ex 5.3 Q5(c)
    having slope 12\dfrac{1}{2} and containing the point (3,−2)(3,-2).
  13. Ex 5.3 Q5(d)
    containing the point A(3,5)A(3,5) and having slope 23\dfrac{2}{3}.
  14. Ex 5.3 Q5(e)
    containing the point A(4,3)A(4,3) and having inclination 120∘120^\circ.
  15. Ex 5.3 Q5(f)
    passing through the origin and which bisects the portion of the line 3x+y=63x + y = 6 intercepted between the co-ordinate axes.
  16. Ex 5.3 Q6
    Line y=mx+cy = mx + c passes through points A(2,1)A(2,1) and B(3,2)B(3,2). Determine mm and cc.
  17. Ex 5.3 Q7
    Find the equation of the line having inclination 135∘135^\circ and making X-intercept 7.
  18. The vertices of a triangle are A(3,4)A(3,4), B(2,0)B(2,0) and C(−1,6)C(-1,6). Find the equations of the lines containing
    Ex 5.3 Q8(a)
    side BC
  19. Ex 5.3 Q8(b)
    the median AD
  20. Ex 5.3 Q8(c)
    the mid points of sides AB and BC.
  21. Find the xx and yy intercepts of the following lines :
    Ex 5.3 Q9(a)
    x3+y2=1\dfrac{x}{3} + \dfrac{y}{2} = 1
  22. Ex 5.3 Q9(b)
    3x2+2y3=1\dfrac{3x}{2} + \dfrac{2y}{3} = 1
  23. Ex 5.3 Q9(c)
    2x−3y+12=02x - 3y + 12 = 0
  24. Ex 5.3 Q10
    Find equations of lines which contains the point A(1,3)A(1,3) and the sum of whose intercepts on the co-ordinate axes is zero.
  25. Ex 5.3 Q11
    Find equations of lines containing the point A(3,4) and making equal intercepts on the co-ordinates axes.
  26. Ex 5.3 Q12
    Find equations of altitudes of the triangle whose vertices are A(2,5)A(2,5), B(6,−1)B(6,-1) and C(−4,−3)C(-4,-3).
  27. Ex 5.3 Q13
    Find the equations of perpendicular bisectors of sides of the triangle whose vertices are P(−1,8)P(-1,8), Q(4,−2)Q(4,-2) and R(−5,−3)R(-5,-3).
  28. Ex 5.3 Q14
    Find the co-ordinates of the orthocenter of the triangle whose vertices are A(2,−2)A(2,-2), B(1,1)B(1,1) and C(−1,0)C(-1,0).
  29. Ex 5.3 Q15
    N(3,−4)N(3,-4) is the foot of the perpendicular drawn from the origin to line L. Find the equation of line L.

5.4 General Form of the Equation of a Line

14 q

Solved Examples

Worked · 14
  1. Find the slope and intercepts made by the following lines :
    5.4 SolvedEx.1(a)
    x+y+10=0x + y + 10 = 0
  2. 5.4 SolvedEx.1(b)
    2x+y+30=02x + y + 30 = 0
  3. 5.4 SolvedEx.1(c)
    x+3y−15=0x + 3y - 15 = 0
  4. Find the acute angle between the following pairs of lines :
    5.4 SolvedEx.2(a)
    12x−4y=512x - 4y = 5 and 4x+2y=74x + 2y = 7
  5. 5.4 SolvedEx.2(b)
    y=2x+3y = 2x + 3 and y=3x+7y = 3x + 7
  6. 5.4 SolvedEx.3
    Find the acute angle between the lines y−3 x+1=0y - \sqrt{3}\,x + 1 = 0 and 3 y−x+7=0\sqrt{3}\,y - x + 7 = 0.
  7. Show that following pairs of lines are perpendicular to each other.
    5.4 SolvedEx.4(a)
    2x−4y=52x - 4y = 5 and 2x+y=172x + y = 17.
  8. 5.4 SolvedEx.4(b)
    y=2x+23y = 2x + 23 and 2x+4y=272x + 4y = 27
  9. 5.4 SolvedEx.5
    Find equations of lines which pass through the origin and make an angle of 45∘45^\circ with the line 3x−y=63x - y = 6.
  10. 5.4 SolvedEx.6
    A line is parallel to the line 2x+y=72x + y = 7 and passes through the origin. Find its equation.
  11. 5.4 SolvedEx.7
    A line is parallel to the line x+3y=9x + 3y = 9 and passes through the point A(2,7)A(2,7). Find its equation.
  12. 5.4 SolvedEx.8
    A line is perpendicular to the line 3x+2y−1=03x + 2y - 1 = 0 and passes through the point A(1,1)A(1,1). Find its equation.
  13. 5.4 SolvedEx.9
    Find the co-ordinates of the point of intersection of lines x+2y=3x + 2y = 3 and 2x−y=12x - y = 1.
  14. 5.4 SolvedEx.10
    Find the equation of line which is parallel to the X-axis and which passes through the point of intersection of lines x+2y=6x + 2y = 6 and 2x−y=22x - y = 2.

5.4.3 Distance Between Two Parallel Lines

3 q

Solved Examples

Worked · 3
  1. 5.4.3 SolvedEx.1
    Find the distance of the origin from the line 3x+4y+15=03x+4y+15=0
  2. 5.4.3 SolvedEx.2
    Find the distance of the point P(2,5)P(2,5) from the line 3x+4y+14=03x+4y+14=0
  3. 5.4.3 SolvedEx.3
    Find the distance between the parallel lines 6x+8y+21=06x+8y+21=0 and 3x+4y+7=03x+4y+7=0.

5.4.4 Family of Lines

2 q

Solved Examples

Worked · 2
  1. 5.4.4 SolvedEx.1
    Find the equation of the line which passes through the point of intersection of lines x+2y+6=0x+2y+6=0, 2x−y=22x-y=2 and which makes intercept 5 on the Y-axis.
  2. 5.4.4 SolvedEx.2
    Find the equation of line which passes through the point of intersection of lines 3x+2y−6=03x+2y-6=0, x+y+1=0x+y+1=0 and the point A(2,1)A(2,1).

Exercise 5.4

26 q
  1. Find the slope, X-intercept, Y-intercept of each of the following lines.
    Ex 5.4 Q1(a)
    2x+3y−6=02x+3y-6=0
  2. Ex 5.4 Q1(b)
    3x−y−9=03x-y-9=0
  3. Ex 5.4 Q1(c)
    x+2y=0x+2y=0
  4. Write each of the following equations in ax+by+c=0ax+by+c=0 form.
    Ex 5.4 Q2(a)
    y=2x−4y=2x-4
  5. Ex 5.4 Q2(b)
    y=4y=4
  6. Ex 5.4 Q2(c)
    x2+y4=1\frac{x}{2}+\frac{y}{4}=1
  7. Ex 5.4 Q2(d)
    x3−y2=0\frac{x}{3}-\frac{y}{2}=0
  8. Ex 5.4 Q3
    Show that lines x−2y−7=0x-2y-7=0 and 2x−4y+15=02x-4y+15=0 are parallel to each other.
  9. Ex 5.4 Q4
    Show that lines x−2y−7=0x-2y-7=0 and 2x+y+1=02x+y+1=0 are perpendicular to each other. Find their point of intersection.
  10. Ex 5.4 Q5
    If the line 3x+4y=p3x+4y=p makes a triangle of area 24 square unit with the co-ordinate axes then find the value of pp.
  11. Ex 5.4 Q6
    Find the co-ordinates of the foot of the perpendicular drawn from the point A(−2,3)A(-2,3) to the line 3x−y−1=03x-y-1=0.
  12. Ex 5.4 Q7
    Find the co-ordinates of the circumcenter of the triangle whose vertices are A(−2,3)A(-2,3), B(6,−1)B(6,-1), C(4,3)C(4,3).
  13. Ex 5.4 Q8
    Find the co-ordinates of the orthocenter of the triangle whose vertices are A(3,−2)A(3,-2), B(7,6)B(7,6), C(−1,2)C(-1,2).
  14. Ex 5.4 Q9
    Show that lines 3x−4y+5=03x-4y+5=0, 7x−8y+5=07x-8y+5=0, and 4x+5y−45=04x+5y-45=0 are concurrent. Find their point of concurrence.
  15. Ex 5.4 Q10
    Find the equation of the line whose X-intercept is 3 and which is perpendicular to the line 3x−y+23=03x-y+23=0.
  16. Ex 5.4 Q11
    Find the distance of the origin from the line 7x+24y−50=07x+24y-50=0.
  17. Ex 5.4 Q12
    Find the distance of the point A(−2,3)A(-2,3) from the line 12x−5y−13=012x-5y-13=0.
  18. Ex 5.4 Q13
    Find the distance between parallel lines 4x−3y+5=04x-3y+5=0 and 4x−3y+7=04x-3y+7=0
  19. Ex 5.4 Q14
    Find the distance between parallel lines 9x+6y−7=09x+6y-7=0 and 3x+2y+6=03x+2y+6=0
  20. Ex 5.4 Q15
    Find points on the line x+y−4=0x+y-4=0 which are at one unit distance from the line x+y−2=0x+y-2=0
  21. Ex 5.4 Q16
    Find the equation of the line parallel to the X-axis and passing through the point of intersection of lines x+y−2=0x+y-2=0 and 4x+3y=104x+3y=10.
  22. Ex 5.4 Q17
    Find the equation of the line passing through the point of intersection of lines x+y−2=0x+y-2=0 and 2x−3y+4=02x-3y+4=0 and making intercept 3 on the X-axis.
  23. Ex 5.4 Q18
    If A(4,3)A(4,3), B(0,0)B(0,0), and C(2,3)C(2,3) are the vertices of △ABC\triangle ABC then find the equation of bisector of angle BAC.
  24. D(−1,8)D(-1,8), E(4,−2)E(4,-2), F(−5,−3)F(-5,-3) are midpoints of sides BC, CA and AB of △ABC\triangle ABC. Find
    Ex 5.4 Q19(i)
    equations of sides of △ABC\triangle ABC.
  25. Ex 5.4 Q19(ii)
    co-ordinates of the circumcenter of △ABC\triangle ABC.
  26. Ex 5.4 Q20
    O(0,0)O(0,0), A(6,0)A(6,0) and B(0,8)B(0,8) are vertices of a triangle. Find the co-ordinates of the incenter of △OAB\triangle OAB.

Miscellaneous Exercise 5

53 q

(I) Select the correct option

Practice · 10
  1. Misc I Q1
    If AA is (5,−3)(5,-3) and BB is a point on the xx-axis such that the slope of line ABAB is −2-2 then B≡B \equiv
    1. A.
      (7,2)(7,2)
    2. B.
      (72,0)\left(\frac{7}{2},0\right)
    3. C.
      (0,72)\left(0,\frac{7}{2}\right)
    4. D.
      (27,0)\left(\frac{2}{7},0\right)
  2. Misc I Q2
    If the point (1,1)(1,1) lies on the line passing through the points (a,0)(a,0) and (0,b)(0,b), then 1a+1b=\frac{1}{a}+\frac{1}{b} =
    1. A.
      −1-1
    2. B.
      00
    3. C.
      11
    4. D.
      1ab\frac{1}{ab}
  3. Misc I Q3
    If A(1,−2)A(1,-2), B(−2,3)B(-2,3) and C(2,−5)C(2,-5) are the vertices of ΔABC\Delta ABC, then the equation of the median BEBE is
    1. A.
      7x+13y+47=07x+13y+47=0
    2. B.
      13x+7y+5=013x+7y+5=0
    3. C.
      7x−13y+5=07x-13y+5=0
    4. D.
      13x−7y−5=013x-7y-5=0
  4. Misc I Q4
    The equation of the line through (1,2)(1,2), which makes equal intercepts on the axes, is
    1. A.
      x+y=1x+y=1
    2. B.
      x+y=2x+y=2
    3. C.
      x+y=4x+y=4
    4. D.
      x+y=3x+y=3
  5. Misc I Q5
    If the line kx+4y=6kx+4y=6 passes through the point of intersection of the two lines 2x+3y=42x+3y=4 and 3x+4y=53x+4y=5, then k=k =
    1. A.
      11
    2. B.
      22
    3. C.
      33
    4. D.
      44
  6. Misc I Q6
    The equation of a line, having inclination 120∘120^\circ with positive direction of XX-axis, which is at a distance of 3 units from the origin is
    1. A.
      3x±y+6=0\sqrt{3}x \pm y+6=0
    2. B.
      3x+y±6=0\sqrt{3}x+y \pm 6=0
    3. C.
      x+y=6x+y=6
    4. D.
      x+y=−6x+y=-6
  7. Misc I Q7
    A line passes through (2,2)(2,2) and is perpendicular to the line 3x+y=33x+y=3. Its yy-intercept is
    1. A.
      13\frac{1}{3}
    2. B.
      23\frac{2}{3}
    3. C.
      11
    4. D.
      43\frac{4}{3}
  8. Misc I Q8
    The angle between the line 3x−y−2=0\sqrt{3}x-y-2=0 and x−3y+1=0x-\sqrt{3}y+1=0 is
    1. A.
      15∘15^\circ
    2. B.
      30∘30^\circ
    3. C.
      45∘45^\circ
    4. D.
      60∘60^\circ
  9. Misc I Q9
    If kx+2y−1=0kx+2y-1=0 and 6x−4y+2=06x-4y+2=0 are identical lines, then determine kk.
    1. A.
      −3-3
    2. B.
      −13-\frac{1}{3}
    3. C.
      13\frac{1}{3}
    4. D.
      33
  10. Misc I Q10
    Distance between the two parallel lines y=2x+7y=2x+7 and y=2x+5y=2x+5 is
    1. A.
      25\frac{\sqrt{2}}{\sqrt{5}}
    2. B.
      15\frac{1}{\sqrt{5}}
    3. C.
      52\frac{\sqrt{5}}{2}
    4. D.
      25\frac{2}{\sqrt{5}}

(II) Answer the following

Practice · 43
  1. Find the value of kk
    Misc II Q1(a)
    if the slope of the line passing through the points P(3,4)P(3,4), Q(5,k)Q(5,k) is 9.
  2. Misc II Q1(b)
    the points A(1,3),B(4,1),C(3,k)A(1,3), B(4,1), C(3,k) are collinear
  3. Misc II Q1(c)
    the point P(1,k)P(1,k) lies on the line passing through the points A(2,2)A(2,2) and B(3,3)B(3,3).
  4. Misc II Q2
    Reduce the equation 6x+3y+8=06x+3y+8=0 into slope-intercept form. Hence find its slope.
  5. Misc II Q3
    Find the distance of the origin from the line x=−2x=-2.
  6. Misc II Q4
    Does point A(2,3)A(2,3) lie on the line 3x+2y−6=03x+2y-6=0? Give reason.
  7. Misc II Q5
    Which of the following lines passes through the origin? (a) x=2x=2 (b) y=3y=3 (c) y=x+2y=x+2 (d) 2x−y=02x-y=0
  8. Obtain the equation of the line which is :
    Misc II Q6(a)
    parallel to the XX-axis and 3 unit below it.
  9. Misc II Q6(b)
    parallel to the YY-axis and 2 unit to the left of it.
  10. Misc II Q6(c)
    parallel to the XX-axis and making an intercept of 5 on the YY-axis.
  11. Misc II Q6(d)
    parallel to the YY-axis and making an intercept of 3 on the XX-axis.
  12. Obtain the equation of the line containing the point
    Misc II Q7(i)
    (2,3)(2,3) and parallel to the XX-axis.
  13. Misc II Q7(ii)
    (2,4)(2,4) and perpendicular to the YY-axis.
  14. Find the equation of the line :
    Misc II Q8(a)
    having slope 5 and containing point A(−1,2)A(-1,2).
  15. Misc II Q8(b)
    containing the point T(7,3)T(7,3) and having inclination 90∘90^\circ.
  16. Misc II Q8(c)
    through the origin which bisects the portion of the line 3x+2y=23x+2y=2 intercepted between the co-ordinate axes.
  17. Misc II Q9
    Find the equation of the line passing through the points S(2,1)S(2,1) and T(2,3)T(2,3).
  18. Misc II Q10
    Find the distance of the origin from the line 12x+5y+78=012x+5y+78=0.
  19. Misc II Q11
    Find the distance between the parallel lines 3x+4y+3=03x+4y+3=0 and 3x+4y+15=03x+4y+15=0.
  20. Misc II Q12
    Find the equation of the line which contains the point A(3,5)A(3,5) and makes equal intercepts on the co-ordinates axes.
  21. The vertices of a triangle are A(1,4)A(1,4), B(2,3)B(2,3) and C(1,6)C(1,6). Find equations of
    Misc II Q13(a)
    the sides
  22. Misc II Q13(b)
    the medians
  23. Misc II Q13(c)
    Perpendicular bisectors of sides
  24. Misc II Q13(d)
    altitudes of ΔABC\Delta ABC.
  25. Misc II Q14
    Find the equation of the line which passes through the point of intersection of lines x+y−3=0x+y-3=0, 2x−y+1=02x-y+1=0 and which is parallel to XX-axis.
  26. Misc II Q15
    Find the equation of the line which passes through the point of intersection of lines x+y+9=0x+y+9=0, 2x+3y+1=02x+3y+1=0 and which makes XX-intercept 1.
  27. Misc II Q16
    Find the equation of the line through A(−2,3)A(-2,3) and perpendicular to the line through S(1,2)S(1,2) and T(2,5)T(2,5).
  28. Misc II Q17
    Find the XX-intercept of the line whose slope is 3 and which makes intercept 4 on the YY-axis.
  29. Misc II Q18
    Find the distance of P(−1,1)P(-1,1) from the line 12(x+6)=5(y−2)12(x+6)=5(y-2).
  30. Misc II Q19
    Line through A(h,3)A(h,3) and B(4,1)B(4,1) intersect the line 7x−9y−19=07x-9y-19=0 at right angle. Find the value of hh.
  31. Misc II Q20
    Two lines passing through M(2,3)M(2,3) intersect each other at an angle of 45∘45^\circ. If slope of one line is 2, find the equation of the other line.
  32. Misc II Q21
    Find the YY-intercept of the line whose slope is 4 and which has XX intercept 5.
  33. Misc II Q22
    Find the equations of the diagonals of the rectangle whose sides are contained in the lines x=8x=8, x=10x=10, y=11y=11 and y=12y=12.
  34. Misc II Q23
    A(1,4)A(1,4), B(2,3)B(2,3) and C(1,6)C(1,6) are vertices of ΔABC\Delta ABC. Find the equation of the altitude through BB and hence find the co-ordinates of the point where this altitude cuts the side ACAC of ΔABC\Delta ABC.
  35. Misc II Q24
    The vertices of ΔPQR\Delta PQR are P(2,1)P(2,1), Q(−2,3)Q(-2,3) and R(4,5)R(4,5). Find the equation of the median through RR.
  36. Misc II Q25
    A line perpendicular to segment joining A(1,0)A(1,0) and B(2,3)B(2,3) divides it internally in the ratio 1:2. Find the equation of the line.
  37. Misc II Q26
    Find the co-ordinates of the foot of the perpendicular drawn from the point P(−1,3)P(-1,3) to the line 3x−4y−16=03x-4y-16=0.
  38. Misc II Q27
    Find points on the XX-axis whose distance from the line x3+y4=1\frac{x}{3}+\frac{y}{4}=1 is 4 unit.
  39. Misc II Q28
    The perpendicular from the origin to a line meets it at (−2,9)(-2,9). Find the equation of the line.
  40. Misc II Q29
    P(a,b)P(a,b) is the mid point of a line segment between axes. Show that the equation of the line is xa+yb=2\frac{x}{a}+\frac{y}{b}=2.
  41. Misc II Q30
    Find the distance of the line 4x−y=04x-y=0 from the point P(4,1)P(4,1) measured along the line making an angle of 135∘135^\circ with the positive XX-axis.
  42. Misc II Q31
    Show that there are two lines which pass through A(3,4)A(3,4) and the sum of whose intercepts is zero.
  43. Misc II Q32
    Show that there is only one line which passes through B(5,5)B(5,5) and the sum of whose intercept is zero.