Sign in

Mathematics · Textbook solutions

Pair of Straight Lines

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 145 questions

  1. 4.1 Solved Ex.1
    Find the combined equation of lines x+y−2=0x + y - 2 = 0 and 2x−y+2=02x - y + 2 = 0.
  2. 4.1 Solved Ex.2
    Find the combined equation of lines x−2=0x - 2 = 0 and y+2=0y + 2 = 0.
  3. 4.1 Solved Ex.3
    Find the combined equation of lines x−2y=0x - 2y = 0 and x+y=0x + y = 0.
  4. 4.1 Solved Ex.4
    Find separate equations of lines represented by x2−y2+x+y=0x^2 - y^2 + x + y = 0.
  1. 4.1 Solved (4.2) Ex.1
    Verify that the combined equation of lines 2x+3y=02x + 3y = 0 and x−2y=0x - 2y = 0 is a homogeneous equation of degree two.
  2. 4.1 Solved (4.2) Ex.1b
    Show that lines represented by equation x2−2xy−3y2=0x^2 - 2xy - 3y^2 = 0 are distinct.
  3. 4.1 Solved (4.2) Ex.2
    Show that lines represented by equation x2−6xy+9y2=0x^2 - 6xy + 9y^2 = 0 are coincident.
  4. 4.1 Solved (4.2) Ex.3
    Find the sum and the product of slopes of lines represented by x2+4xy−7y2=0x^2 + 4xy - 7y^2 = 0.
  5. Find the separate equations of lines represented by:
    4.1 Solved (4.2) Ex.4 (i)
    x2−4y2=0x^2 - 4y^2 = 0
  6. 4.1 Solved (4.2) Ex.4 (ii)
    3x2−7xy+4y2=03x^2 - 7xy + 4y^2 = 0
  7. 4.1 Solved (4.2) Ex.4 (iii)
    x2+2xy−y2=0x^2 + 2xy - y^2 = 0
  8. 4.1 Solved (4.2) Ex.4 (iv)
    5x2−3y2=05x^2 - 3y^2 = 0
  9. 4.1 Solved (4.2) Ex.5
    Find the value of kk if 2x+y=02x + y = 0 is one of the lines represented by 3x2+kxy+2y2=03x^2 + kxy + 2y^2 = 0.
  10. 4.1 Solved (4.2) Ex.6
    Find the condition that the line 3x−2y=03x - 2y = 0 coincides with one of the lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0.
  11. 4.1 Solved (4.2) Ex.7
    Find the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by 3x2+2xy−y2=03x^2 + 2xy - y^2 = 0.
  12. 4.1 Solved (4.2) Ex.8
    Find the value of kk, if slope of one of the lines represented by 4x2+kxy+y2=04x^2 + kxy + y^2 = 0 is four times the slope of the other line.
  1. Find the combined equation of the following pairs of lines:
    Q.1 (i)
    2x+y=02x + y = 0 and 3x−y=03x - y = 0
  2. Q.1 (ii)
    x+2y−1=0x + 2y - 1 = 0 and x−3y+2=0x - 3y + 2 = 0
  3. Q.1 (iii)
    Passing through (2,3)(2, 3) and parallel to the co-ordinate axes.
  4. Q.1 (iv)
    Passing through (2,3)(2, 3) and perpendicular to lines 3x+2y−1=03x + 2y - 1 = 0 and x−3y+2=0x - 3y + 2 = 0.
  5. Q.1 (v)
    Passing through (−1,2)(-1, 2), one is parallel to x+3y−1=0x + 3y - 1 = 0 and the other is perpendicular to 2x−3y−1=02x - 3y - 1 = 0.
  6. Find the separate equations of the lines represented by following equations:
    Q.2 (i)
    3y2+7xy=03y^2 + 7xy = 0
  7. Q.2 (ii)
    5x2−9y2=05x^2 - 9y^2 = 0
  8. Q.2 (iii)
    x2−4xy=0x^2 - 4xy = 0
  9. Q.2 (iv)
    3x2−10xy−8y2=03x^2 - 10xy - 8y^2 = 0
  10. Q.2 (v)
    3x2−23 xy−3y2=03x^2 - 2\sqrt{3}\,xy - 3y^2 = 0
  11. Q.2 (vi)
    x2+2(csc⁡α)xy+y2=0x^2 + 2(\csc \alpha)xy + y^2 = 0
  12. Q.2 (vii)
    x2+2xytan⁡α−y2=0x^2 + 2xy \tan \alpha - y^2 = 0
  13. Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by following equations:
    Q.3 (i)
    5x2−8xy+3y2=05x^2 - 8xy + 3y^2 = 0
  14. Q.3 (ii)
    5x2+2xy−3y2=05x^2 + 2xy - 3y^2 = 0
  15. Q.3 (iii)
    xy+y2=0xy + y^2 = 0
  16. Q.3 (iv)
    3x2−4xy=03x^2 - 4xy = 0
  17. Find kk if,
    Q.4 (i)
    the sum of the slopes of the lines represented by x2+kxy−3y2=0x^2 + kxy - 3y^2 = 0 is twice their product.
  18. Q.4 (ii)
    slopes of lines represented by 3x2+kxy−y2=03x^2 + kxy - y^2 = 0 differ by 4.
  19. Q.4 (iii)
    slope of one of the lines given by kx2+4xy−y2=0kx^2 + 4xy - y^2 = 0 exceeds the slope of the other by 8.
  20. Find the condition that:
    Q.5 (i)
    the line 4x+5y=04x + 5y = 0 coincides with one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0.
  21. Q.5 (ii)
    the line 3x+y=03x + y = 0 may be perpendicular to one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0.
  22. Q.6
    If one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is perpendicular to px+qy=0px + qy = 0 then show that ap2+2hpq+bq2=0ap^2 + 2hpq + bq^2 = 0.
  23. Q.7
    Find the combined equation of the pair of lines passing through the origin and making an equilateral triangle with the line y=3y = 3.
  24. Q.8
    If slope of one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is four times the other then show that 16h2=25ab16h^2 = 25ab.
  25. Q.9
    If one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 bisects an angle between co-ordinate axes then show that (a+b)2=4h2(a + b)^2 = 4h^2.
  1. 4.2 Ex.1
    Show that lines represented by 3x2−4xy−3y2=03x^2 - 4xy - 3y^2 = 0 are perpendicular to each other.
  2. 4.2 Ex.2
    Show that lines represented by x2+4xy+4y2=0x^2 + 4xy + 4y^2 = 0 are coincident.
  3. Find the acute angle between lines represented by:
    i)
    x2+xy=0x^2 + xy = 0
  4. ii)
    x2−4xy+y2=0x^2 - 4xy + y^2 = 0
  5. iii)
    3x2+2xy−y2=03x^2 + 2xy - y^2 = 0
  6. iv)
    2x2−6xy+y2=02x^2 - 6xy + y^2 = 0
  7. v)
    xy+y2=0xy + y^2 = 0
  8. 4.2 Ex.4
    Find the combined equation of lines passing through the origin and making angle π6\dfrac{\pi}{6} with the line 3x+y−6=03x + y - 6 = 0.
  9. 4.2 Ex.5
    Find the combined equation of lines passing through the origin and each of which making angle 60∘60^\circ with the X-axis.
  1. 4.2 Q2
    Show that lines represented by x2+6xy+9y2=0x^2 + 6xy + 9y^2 = 0 are coincident.
  2. 4.2 Q3
    Find the value of kk if lines represented by kx2+4xy−4y2=0kx^2 + 4xy - 4y^2 = 0 are perpendicular to each other.
  3. Find the measure of the acute angle between the lines represented by:
    4.2 Q4 i)
    3x2−43 xy+3y2=03x^2 - 4\sqrt{3}\,xy + 3y^2 = 0
  4. 4.2 Q4 ii)
    4x2+5xy+y2=04x^2 + 5xy + y^2 = 0
  5. 4.2 Q4 iii)
    2x2+7xy+3y2=02x^2 + 7xy + 3y^2 = 0
  6. 4.2 Q4 iv)
    (a2−3b2)x2+8abxy+(b2−3a2)y2=0(a^2 - 3b^2)x^2 + 8abxy + (b^2 - 3a^2)y^2 = 0
  7. 4.2 Q5
    Find the combined equation of lines passing through the origin each of which making an angle of 30∘30^\circ with the line 3x+2y−11=03x + 2y - 11 = 0.
  8. 4.2 Q6
    If the angle between lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is equal to the angle between lines represented by 2x2−5xy+3y2=02x^2 - 5xy + 3y^2 = 0 then show that 100(h2−ab)=(a+b)2100(h^2 - ab) = (a + b)^2.
  9. 4.2 Q7
    Find the combined equation of lines passing through the origin and each of which making angle 60∘60^\circ with the Y-axis.
  1. 4.3 Ex.1
    Show that equation x2−6xy+5y2+10x−14y+9=0x^2 - 6xy + 5y^2 + 10x - 14y + 9 = 0 represents a pair of lines. Find the acute angle between them. Also find the point of their intersection.
  2. 4.3 Ex.2
    Find the value of kk if the equation 2x2+4xy−2y2+4x+8y+k=02x^2 + 4xy - 2y^2 + 4x + 8y + k = 0 represents a pair of lines.
  3. 4.3 Ex.3
    Find pp and qq if the equation 2x2+4xy−py2+4x+qy+1=02x^2 + 4xy - py^2 + 4x + qy + 1 = 0 represents a pair of perpendicular lines.
  4. 4.3 Ex.4
    △OAB\triangle OAB is formed by lines x2−4xy+y2=0x^2 - 4xy + y^2 = 0 and the line x+y−2=0x + y - 2 = 0. Find the equation of the median of the triangle drawn from O.
  1. Find the joint equation of the pair of lines:
    4.3 Q1 i)
    Through the point (2,−1)(2, -1) and parallel to lines represented by 2x2+3xy−9y2=02x^2 + 3xy - 9y^2 = 0.
  2. 4.3 Q1 ii)
    Through the point (2,−3)(2, -3) and parallel to lines represented by x2+xy−y2=0x^2 + xy - y^2 = 0.
  3. 4.3 Q2
    Show that equation x2+2xy+2y2+2x+2y+1=0x^2 + 2xy + 2y^2 + 2x + 2y + 1 = 0 does not represent a pair of lines.
  4. 4.3 Q3
    Show that equation 2x2−xy−3y2−6x+19y−20=02x^2 - xy - 3y^2 - 6x + 19y - 20 = 0 represents a pair of lines.
  5. 4.3 Q4
    Show the equation 2x2+xy−y2+x+4y−3=02x^2 + xy - y^2 + x + 4y - 3 = 0 represents a pair of lines. Also find the acute angle between them.
  6. Find the separate equation of the lines represented by the following equations:
    4.3 Q5 i)
    (x−2)2−3(x−2)(y+1)+2(y+1)2=0(x - 2)^2 - 3(x - 2)(y + 1) + 2(y + 1)^2 = 0
  7. 4.3 Q5 ii)
    10(x+1)2+(x+1)(y−2)−3(y−2)2=010(x + 1)^2 + (x + 1)(y - 2) - 3(y - 2)^2 = 0
  8. Find the value of kk if the following equations represent a pair of lines:
    4.3 Q6 i)
    3x2+10xy+3y2+16y+k=03x^2 + 10xy + 3y^2 + 16y + k = 0
  9. 4.3 Q6 ii)
    kxy+10x+6y+4=0kxy + 10x + 6y + 4 = 0
  10. 4.3 Q6 iii)
    x2+3xy+2y2+x−y+k=0x^2 + 3xy + 2y^2 + x - y + k = 0
  11. 4.3 Q7
    Find pp and qq if the equation px2−8xy+3y2+14x+2y+q=0px^2 - 8xy + 3y^2 + 14x + 2y + q = 0 represents a pair of perpendicular lines.
  12. 4.3 Q8
    Find pp and qq if the equation 2x2+8xy+py2+qx+2y−15=02x^2 + 8xy + py^2 + qx + 2y - 15 = 0 represents a pair of parallel lines.
  13. 4.3 Q9
    Equations of pairs of opposite sides of a parallelogram are x2−7x+6=0x^2 - 7x + 6 = 0 and y2−14y+40=0y^2 - 14y + 40 = 0. Find the joint equation of its diagonals.
  14. 4.3 Q10
    △OAB\triangle OAB is formed by lines x2−4xy+y2=0x^2 - 4xy + y^2 = 0 and the line 2x+3y−1=02x + 3y - 1 = 0. Find the equation of the median of the triangle drawn from O.
  15. 4.3 Q11
    Find the co-ordinates of the points of intersection of the lines represented by x2−y2−2x+1=0x^2 - y^2 - 2x + 1 = 0.
  1. Misc I (1)
    If the equation 4x2+hxy+y2=04x^2 + hxy + y^2 = 0 represents two coincident lines, then h=h = __________.
    1. A.
      ±2\pm 2
    2. B.
      ±3\pm 3
    3. C.
      ±4\pm 4
    4. D.
      ±5\pm 5
  2. Misc I (2)
    If the lines represented by kx2−3xy+6y2=0kx^2 - 3xy + 6y^2 = 0 are perpendicular to each other then __________.
    1. A.
      k=6k = 6
    2. B.
      k=−6k = -6
    3. C.
      k=3k = 3
    4. D.
      k=−3k = -3
  3. Misc I (3)
    Auxiliary equation of 2x2+3xy−9y2=02x^2 + 3xy - 9y^2 = 0 is __________.
    1. A.
      2m2+3m−9=02m^2 + 3m - 9 = 0
    2. B.
      9m2−3m−2=09m^2 - 3m - 2 = 0
    3. C.
      2m2−3m+9=02m^2 - 3m + 9 = 0
    4. D.
      −9m2−3m+2=0-9m^2 - 3m + 2 = 0
  4. Misc I (4)
    The difference between the slopes of the lines represented by 3x2−4xy+y2=03x^2 - 4xy + y^2 = 0 is __________.
    1. A.
      2
    2. B.
      1
    3. C.
      3
    4. D.
      4
  5. Misc I (5)
    If the two lines ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 make angles α\alpha and β\beta with X-axis, then tan⁡(α+β)=\tan(\alpha + \beta) = __________.
    1. A.
      ha+b\dfrac{h}{a+b}
    2. B.
      ha−b\dfrac{h}{a-b}
    3. C.
      2ha+b\dfrac{2h}{a+b}
    4. D.
      2ha−b\dfrac{2h}{a-b}
  6. Misc I (6)
    If the slope of one of the two lines x2a+2xyh+y2b=0\dfrac{x^2}{a} + \dfrac{2xy}{h} + \dfrac{y^2}{b} = 0 is twice that of the other, then ab:h2=ab : h^2 = __________.
    1. A.
      1:21 : 2
    2. B.
      2:12 : 1
    3. C.
      8:98 : 9
    4. D.
      9:89 : 8
  7. Misc I (7)
    The joint equation of the lines through the origin and perpendicular to the pair of lines 3x2+4xy−5y2=03x^2 + 4xy - 5y^2 = 0 is __________.
    1. A.
      5x2+4xy−3y2=05x^2 + 4xy - 3y^2 = 0
    2. B.
      3x2+4xy−5y2=03x^2 + 4xy - 5y^2 = 0
    3. C.
      3x2−4xy+5y2=03x^2 - 4xy + 5y^2 = 0
    4. D.
      5x2+4xy+3y2=05x^2 + 4xy + 3y^2 = 0
  8. Misc I (8)
    If acute angle between lines ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is π4\dfrac{\pi}{4}, then 4h2=4h^2 = __________.
    1. A.
      a2+4ab+b2a^2 + 4ab + b^2
    2. B.
      a2+6ab+b2a^2 + 6ab + b^2
    3. C.
      (a+2b)(a+3b)(a + 2b)(a + 3b)
    4. D.
      (a−2b)(2a+b)(a - 2b)(2a + b)
  9. Misc I (9)
    If the equation 3x2−8xy+qy2+2x+14y+p=13x^2 - 8xy + qy^2 + 2x + 14y + p = 1 represents a pair of perpendicular lines then the values of pp and qq are respectively __________.
    1. A.
      −3-3 and −7-7
    2. B.
      −7-7 and −3-3
    3. C.
      3 and 7
    4. D.
      −7-7 and 3
  10. Misc I (10)
    The area of triangle formed by the lines x2+4xy+y2=0x^2 + 4xy + y^2 = 0 and x−y−4=0x - y - 4 = 0 is __________.
    1. A.
      43\dfrac{4}{\sqrt{3}} Sq. units
    2. B.
      83\dfrac{8}{\sqrt{3}} Sq. units
    3. C.
      163\dfrac{16}{\sqrt{3}} Sq. units
    4. D.
      153\dfrac{15}{\sqrt{3}} Sq. units
  11. Misc I (11)
    The combined equation of the co-ordinate axes is __________.
    1. A.
      x+y=0x + y = 0
    2. B.
      xy=kxy = k
    3. C.
      xy=0xy = 0
    4. D.
      x−y=kx - y = k
  12. Misc I (12)
    If h2=abh^2 = ab, then slope of lines ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 are in the ratio __________.
    1. A.
      1:21 : 2
    2. B.
      2:12 : 1
    3. C.
      2:32 : 3
    4. D.
      1:11 : 1
  13. Misc I (13)
    If slope of one of the lines ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is 5 times the slope of the other, then 5h2=5h^2 = __________.
    1. A.
      abab
    2. B.
      2ab2ab
    3. C.
      7ab7ab
    4. D.
      9ab9ab
  14. Misc I (14)
    If distance between lines (x−2y)2+k(x−2y)=0(x - 2y)^2 + k(x - 2y) = 0 is 3 units, then k=k = __________.
    1. A.
      ±3\pm 3
    2. B.
      ±55\pm 5\sqrt{5}
    3. C.
      0
    4. D.
      ±35\pm 3\sqrt{5}
  1. Find the joint equation of lines:
    Misc II Q.1 (i)
    x−y=0x - y = 0 and x+y=0x + y = 0
  2. Misc II Q.1 (ii)
    x+y−3=0x + y - 3 = 0 and 2x+y−1=02x + y - 1 = 0
  3. Misc II Q.1 (iii)
    Passing through the origin and having slopes 2 and 3.
  4. Misc II Q.1 (iv)
    Passing through the origin and having inclinations 60∘60^\circ and 120∘120^\circ.
  5. Misc II Q.1 (v)
    Passing through (1,2)(1, 2) and parallel to the co-ordinate axes.
  6. Misc II Q.1 (vi)
    Passing through (3,2)(3, 2) and parallel to the line x=2x = 2 and y=3y = 3.
  7. Misc II Q.1 (vii)
    Passing through (−1,2)(-1, 2) and perpendicular to the lines x+2y+3=0x + 2y + 3 = 0 and 3x−4y−5=03x - 4y - 5 = 0.
  8. Misc II Q.1 (viii)
    Passing through the origin and having slopes 1+31 + \sqrt{3} and 1−31 - \sqrt{3}.
  9. Misc II Q.1 (ix)
    Which are at a distance of 9 units from the Y-axis.
  10. Misc II Q.1 (x)
    Passing through the point (3,2)(3, 2), one of which is parallel to the line x−2y=2x - 2y = 2 and other is perpendicular to the line y=3y = 3.
  11. Misc II Q.1 (xi)
    Passing through the origin and perpendicular to the lines x+2y=19x + 2y = 19 and 3x+y=183x + y = 18.
  12. Show that each of the following equation represents a pair of lines.
    Misc II Q.2 (i)
    x2+2xy−y2=0x^2 + 2xy - y^2 = 0
  13. Misc II Q.2 (ii)
    4x2+4xy+y2=04x^2 + 4xy + y^2 = 0
  14. Misc II Q.2 (iii)
    x2−y2=0x^2 - y^2 = 0
  15. Misc II Q.2 (iv)
    x2+7xy−2y2=0x^2 + 7xy - 2y^2 = 0
  16. Misc II Q.2 (v)
    x2−23 xy−y2=0x^2 - 2\sqrt{3}\,xy - y^2 = 0
  17. Find the separate equations of lines represented by the following equations:
    Misc II Q.3 (i)
    6x2−5xy−6y2=06x^2 - 5xy - 6y^2 = 0
  18. Misc II Q.3 (ii)
    x2−4y2=0x^2 - 4y^2 = 0
  19. Misc II Q.3 (iii)
    3x2−y2=03x^2 - y^2 = 0
  20. Misc II Q.3 (iv)
    2x2+2xy−y2=02x^2 + 2xy - y^2 = 0
  21. Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by:
    Misc II Q.4 (i)
    x2+4xy−5y2=0x^2 + 4xy - 5y^2 = 0
  22. Misc II Q.4 (ii)
    2x2−3xy−9y2=02x^2 - 3xy - 9y^2 = 0
  23. Misc II Q.4 (iii)
    x2+xy−y2=0x^2 + xy - y^2 = 0
  24. Find kk if:
    Misc II Q.5 (i)
    The sum of the slopes of the lines given by 3x2+kxy−y2=03x^2 + kxy - y^2 = 0 is zero.
  25. Misc II Q.5 (ii)
    The sum of slopes of the lines given by 2x2+kxy−3y2=02x^2 + kxy - 3y^2 = 0 is equal to their product.
  26. Misc II Q.5 (iii)
    The slope of one of the lines given by 3x2−4xy+ky2=03x^2 - 4xy + ky^2 = 0 is 1.
  27. Misc II Q.5 (iv)
    One of the lines given by 3x2−kxy+5y2=03x^2 - kxy + 5y^2 = 0 is perpendicular to the 5x+3y=05x + 3y = 0.
  28. Misc II Q.5 (v)
    The slope of one of the lines given by 3x2+4xy+ky2=03x^2 + 4xy + ky^2 = 0 is three times the other.
  29. Misc II Q.5 (vi)
    The slopes of lines given by kx2+5xy+y2=0kx^2 + 5xy + y^2 = 0 differ by 1.
  30. Misc II Q.5 (vii)
    One of the lines given by 6x2+kxy+y2=06x^2 + kxy + y^2 = 0 is 2x+y=02x + y = 0.
  31. Misc II Q.6
    Find the joint equation of the pair of lines which bisect angle between the lines given by x2+3xy+2y2=0x^2 + 3xy + 2y^2 = 0.
  32. Misc II Q.7
    Find the joint equation of the pair of lines through the origin and making equilateral triangle with the line x=3x = 3.
  33. Misc II Q.8
    Show that the lines x2−4xy+y2=0x^2 - 4xy + y^2 = 0 and x+y=10x + y = 10 contain the sides of an equilateral triangle. Find the area of the triangle.
  34. Misc II Q.9
    If the slope of one of the lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is three times the other then prove that 3h2=4ab3h^2 = 4ab.
  35. Misc II Q.10
    Find the combined equation of the bisectors of the angles between the lines represented by 5x2+6xy−y2=05x^2 + 6xy - y^2 = 0.
  36. Misc II Q.11
    Find aa if the sum of slope of lines represented by ax2+8xy+5y2=0ax^2 + 8xy + 5y^2 = 0 is twice their product.
  37. Misc II Q.12
    If line 4x−5y=04x - 5y = 0 coincides with one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 then show that 25a+40h+16b=025a + 40h + 16b = 0.
  38. Show that the following equations represent a pair of lines, find the acute angle between them.
    Misc II Q.13 (i)
    9x2−6xy+y2+18x−6y+8=09x^2 - 6xy + y^2 + 18x - 6y + 8 = 0
  39. Misc II Q.13 (ii)
    2x2+xy−y2+x+4y−3=02x^2 + xy - y^2 + x + 4y - 3 = 0
  40. Misc II Q.13 (iii)
    (x−3)2+(x−3)(y−4)−2(y−4)2=0(x - 3)^2 + (x - 3)(y - 4) - 2(y - 4)^2 = 0
  41. Misc II Q.14
    Find the combined equation of pair of lines through the origin each of which makes angle of 60∘60^\circ with the Y-axis.
  42. Misc II Q.15
    If lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 make angles of equal measures with the co-ordinate axes then show that a=±ba = \pm b.
  43. Misc II Q.16
    Show that the combined equation of a pair of lines through the origin and each making an angle of α\alpha with the line x+y=0x + y = 0 is x2+2(sec⁡2α) xy+y2=0x^2 + 2(\sec 2\alpha)\,xy + y^2 = 0.
  44. Misc II Q.17
    Show that the line 3x+4y+5=03x + 4y + 5 = 0 and the lines (3x+4y)2−3(4x−3y)2=0(3x + 4y)^2 - 3(4x - 3y)^2 = 0 form an equilateral triangle.
  45. Misc II Q.18
    Show that lines x2−4xy+y2=0x^2 - 4xy + y^2 = 0 and x+y=6x + y = \sqrt{6} form an equilateral triangle. Find its area and perimeter.
  46. Misc II Q.19
    If the slope of one of the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is square of the other then show that a2b+ab2+8h3=6abha^2 b + ab^2 + 8h^3 = 6abh.
  47. Misc II Q.20
    Prove that the product of lengths of perpendiculars drawn from P(x1,y1)P(x_1, y_1) to the lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is ∣ax12+2hx1y1+by12(a−b)2+4h2∣\left|\dfrac{ax_1^2 + 2hx_1 y_1 + by_1^2}{\sqrt{(a-b)^2 + 4h^2}}\right|.
  48. Misc II Q.21
    Show that the difference between the slopes of lines given by (tan⁡2θ+cos⁡2θ)x2−2xytan⁡θ+(sin⁡2θ)y2=0(\tan^2\theta + \cos^2\theta)x^2 - 2xy\tan\theta + (\sin^2\theta)y^2 = 0 is two.
  49. Misc II Q.22
    Find the condition that the equation ay2+bxy+ex+dy=0ay^2 + bxy + ex + dy = 0 may represent a pair of lines.
  50. Misc II Q.23
    If the lines given by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 form an equilateral triangle with the line lx+my=1lx + my = 1 then show that (3a+b)(a+3b)=4h2(3a + b)(a + 3b) = 4h^2.
  51. Misc II Q.24
    If line x+2=0x + 2 = 0 coincides with one of the lines represented by the equation x2+2xy+4y+k=0x^2 + 2xy + 4y + k = 0 then show that k=−4k = -4.
  52. Misc II Q.25
    Prove that the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by ax2+2hxy+by2=0ax^2 + 2hxy + by^2 = 0 is bx2−2hxy+ay2=0bx^2 - 2hxy + ay^2 = 0.
  53. Misc II Q.26
    If equation ax2−y2+2y+c=1ax^2 - y^2 + 2y + c = 1 represents a pair of perpendicular lines then find aa and cc.