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Mathematics · Class 12 Science

Ch 4Pair of Straight Lines — Class 12 Mathematics, concept-first.

We already know that an equation of the form (with and not both zero) describes a single straight line in the -plane. This section asks what happens when we want a single equation to describe two lines at once.

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Combined Equation of a Pair of Lines

If and are two lines, their product is called the combined (or joint) equation of the pair: it is a single equation whose solution set is exactly the union of the two lines, because a product of real numbers is zero prec…

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4.1

Combined equation of a pair of lines

We already know that an equation of the form (with and not both zero) describes a single straight line in the -plane.

+Exercise 4.1i25 questions
  1. Q1Find the combined equation of the following pair of lines: $2x + y = 0$ and $3x - y = 0$.Free
  2. Q2Find the combined equation of the following pair of lines: $x + 2y - 1 = 0$ and $x - 3y + 2 = 0$.Free
  3. Q3Find the combined equation of the following pair of lines: passing through $(2,3)$ and parallel to the co-ordinate axes.Free
  4. Q4Find the combined equation of the following pair of lines: passing through $(2,3)$ and perpendicular to lines $3x + 2y - 1 = 0$ and $x - 3y…Preview
  5. Q5Find the combined equation of the following pair of lines: passing through $(-1,2)$, one is parallel to $x + 3y - 1 = 0$ and the other is pe…Preview
  6. Q6Find the separate equations of the lines represented by the following equation: $3y^2 + 7xy = 0$.Preview
  7. Q7Find the separate equations of the lines represented by the following equation: $5x^2 - 9y^2 = 0$.Preview
  8. Q8Find the separate equations of the lines represented by the following equation: $x^2 - 4xy = 0$.Preview
  9. Q9Find the separate equations of the lines represented by the following equation: $3x^2 - 10xy - 8y^2 = 0$.Preview
  10. Q10Find the separate equations of the lines represented by the following equation: $3x^2 - 2\sqrt{3}xy - 3y^2 = 0$.Preview
  11. Q11Find the separate equations of the lines represented by the following equation: $x^2 + 2(\text{cosec}\,\alpha)xy + y^2 = 0$.Preview
  12. Q12Find the separate equations of the lines represented by the following equation: $x^2 + 2xy\tan\alpha - y^2 = 0$.Preview
  13. Q13Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by the following equatio…Preview
  14. Q14Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by the following equatio…Preview
  15. Q15Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by the following equatio…Preview
  16. Q16Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by the following equatio…Preview
  17. Q17Find $k$ if the sum of the slopes of the lines represented by $x^2 + kxy - 3y^2 = 0$ is twice their product.Preview
  18. Q18Find $k$ if slopes of lines represented by $3x^2 + kxy - y^2 = 0$ differ by $4$.Preview
  19. Q19Find $k$ if slope of one of the lines given by $kx^2 + 4xy - y^2 = 0$ exceeds the slope of the other by $8$.Preview
  20. Q20Find the condition that the line $4x + 5y = 0$ coincides with one of the lines given by $ax^2 + 2hxy + by^2 = 0$.Preview
  21. Q21Find the condition that the line $3x + y = 0$ may be perpendicular to one of the lines given by $ax^2 + 2hxy + by^2 = 0$.Preview
  22. Q22If one of the lines given by $ax^2 + 2hxy + by^2 = 0$ is perpendicular to $px + qy = 0$ then show that $ap^2 + 2hpq + bq^2 = 0$.Preview
  23. Q23Find the combined equation of the pair of lines passing through the origin and making an equilateral triangle with the line $y = 3$.Preview
  24. Q24If slope of one of the lines given by $ax^2 + 2hxy + by^2 = 0$ is four times the other then show that $16h^2 = 25ab$.Preview
  25. Q25If one of the lines given by $ax^2 + 2hxy + by^2 = 0$ bisects an angle between co-ordinate axes then show that $(a+b)^2 = 4h^2$.Preview
4.2

Homogeneous equation of degree two

A homogeneous equation of degree two in and has the general form . This section shows exactly when such an equation is the combined equation of a pair of lines through the origin, and how to recover t…

+Exercise 4.2i10 questions
  1. Q26Show that lines represented by $3x^2 - 4xy - 3y^2 = 0$ are perpendicular to each other.Free
  2. Q27Show that lines represented by $x^2 + 6xy + 9y^2 = 0$ are coincident.Free
  3. Q28Find the value of $k$ if lines represented by $kx^2 + 4xy - 4y^2 = 0$ are perpendicular to each other.Free
  4. Q29Find the measure of the acute angle between the lines represented by: $3x^2 - 4\sqrt{3}xy + 3y^2 = 0$.Preview
  5. Q30Find the measure of the acute angle between the lines represented by: $4x^2 + 5xy + y^2 = 0$.Preview
  6. Q31Find the measure of the acute angle between the lines represented by: $2x^2 + 7xy + 3y^2 = 0$.Preview
  7. Q32Find the measure of the acute angle between the lines represented by: $(a^2 - 3b^2)x^2 + 8abxy + (b^2 - 3a^2)y^2 = 0$.Preview
  8. Q33Find the combined equation of lines passing through the origin each of which making an angle of $30°$ with the line $3x + 2y - 11 = 0$.Preview
  9. Q34If the angle between lines represented by $ax^2 + 2hxy + by^2 = 0$ is equal to the angle between lines represented by $2x^2 - 5xy + 3y^2 = 0…Preview
  10. Q35Find the combined equation of lines passing through the origin and each of which making angle $60°$ with the Y-axis.Preview
4.2.1

Degree of a term

The degree of a term in an algebraic expression is the sum of the exponents (indices) of all the variables appearing in that term.

4.2.2

Homogeneous Equation

An equation in which every term has the same degree is called a homogeneous equation. For instance , , and are all homogeneous, because in each case every term listed has degree two.

4.3

Angle between lines represented by $ax^2 + 2hxy + by^2 = 0$

This section asks: given the homogeneous pair , what is the angle between its two lines?

+Exercise 4.3i15 questions
  1. Q36Find the joint equation of the pair of lines through the point $(2, -1)$ and parallel to lines represented by $2x^2 + 3xy - 9y^2 = 0$.Free
  2. Q37Find the joint equation of the pair of lines through the point $(2, -3)$ and parallel to lines represented by $x^2 + xy - y^2 = 0$.Free
  3. Q38Show that equation $x^2 + 2xy + 2y^2 + 2x + 2y + 1 = 0$ does not represent a pair of lines.Free
  4. Q39Show that equation $2x^2 - xy - 3y^2 - 6x + 19y - 20 = 0$ represents a pair of lines.Preview
  5. Q40Show the equation $2x^2 + xy - y^2 + x + 4y - 3 = 0$ represents a pair of lines. Also find the acute angle between them.Preview
  6. Q41Find the separate equation of the lines represented by the following equation: $(x-2)^2 - 3(x-2)(y+1) + 2(y+1)^2 = 0$.Preview
  7. Q42Find the separate equation of the lines represented by the following equation: $10(x+1)^2 + (x+1)(y-2) - 3(y-2)^2 = 0$.Preview
  8. Q43Find the value of $k$ if the following equation represents a pair of lines: $3x^2 + 10xy + 3y^2 + 16y + k = 0$.Preview
  9. Q44Find the value of $k$ if the following equation represents a pair of lines: $kxy + 10x + 6y + 4 = 0$.Preview
  10. Q45Find the value of $k$ if the following equation represents a pair of lines: $x^2 + 3xy + 2y^2 + x - y + k = 0$.Preview
  11. Q46Find $p$ and $q$ if the equation $px^2 - 8xy + 3y^2 + 14x + 2y + q = 0$ represents a pair of perpendicular lines.Preview
  12. Q47Find $p$ and $q$ if the equation $2x^2 + 8xy + py^2 + qx + 2y - 15 = 0$ represents a pair of parallel lines.Preview
  13. Q48Equations of pairs of opposite sides of a parallelogram are $x^2 - 7x + 6 = 0$ and $y^2 - 14y + 40 = 0$. Find the joint equation of its diag…Preview
  14. Q49$\triangle OAB$ is formed by lines $x^2 - 4xy + y^2 = 0$ and the line $2x + 3y - 1 = 0$. Find the equation of the median of the triangle dra…Preview
  15. Q50Find the co-ordinates of the points of intersection of the lines represented by $x^2 - y^2 - 2x + 1 = 0$.Preview
4.4

General Second Degree Equation in x and y

So far every pair of lines considered has passed through the origin (via a homogeneous equation) or been built directly from two given linear factors.

4.4.1

The necessary conditions for a general second degree equation

For the general second-degree equation to represent a genuine pair of straight lines, two conditions must both hold:

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 22 questions22 questions
  1. Q1The joint equation of the pair of lines passing through $(2, 3)$ and parallel to the coordinate axes is (a) $xy - 3x - 2y + 6 = 0$ (b) $xy +…Preview
  2. Q2Find $k$, if one of the lines given by $6x^2 + kxy + y^2 = 0$ is $2x + y = 0$.Preview
  3. Q3Show that every homogeneous equation of degree two in $x$ and $y$, i.e., $ax^2 + 2hxy + by^2 = 0$ represents a pair of lines passing through…Preview
  4. Q4If $\theta$ is the measure of the acute angle between the lines represented by the equation $ax^2 + 2hxy + by^2 = 0$, then prove that $\tan\…Preview
  5. Q5Find the joint equation of the pair of lines passing through the origin which are perpendicular respectively to the lines represented by $5x…Preview
  6. Q6Show that a homogeneous equation of degree two in $x$ and $y$, i.e. $ax^2 + 2hxy + by^2 = 0$ represents a pair of lines passing through the…Preview
  7. Q7Find the joint equation of the pair of lines passing through the origin, which are perpendicular to the lines represented by $5x^2 + 2xy - 3…Preview
  8. Q8Find 'p' and 'q' if the equation $px^2 - 8xy + 3y^2 + 14x + 2y + q = 0$ represents a pair of perpendicular lines.Preview
  9. Q9Find the angle between the lines represented by $3x^2+4xy-3y^2=0$Preview
  10. Q10Prove that a homogeneous equation of degree two in $x$ and $y$ i.e. $ax^2+2hxy+by^2=0$ represents a pair of lines passing through the origin…Preview
  11. Q11Write the separate equations of lines represented by the equation $5x^2 - 9y^2 = 0$Preview
  12. Q12Find the value of $k$, if $2x+y=0$ is one of the lines represented by $3x^2+kxy+2y^2=0$Preview
  13. Q13If angle between the lines represented by $ax^2+2hxy+by^2=0$ is equal to the angle between the lines represented by $2x^2-5xy+3y^2=0$, then…Preview
  14. Q14Write the joint equation of co-ordinate axes.Preview
  15. Q15If $ax^2+2hxy+by^2=0$ represents a pair of lines and $h^2=ab\ne 0$ then find the ratio of their slopes.Preview
  16. Q16If $\theta$ is the acute angle between the lines represented by $ax^2+2hxy+by^2=0$ then prove that $\tan\theta = \left|\dfrac{2\sqrt{h^2-ab}…Preview
  17. Q17Find $k$, if the sum of the slopes of the lines represented by $x^2+kxy-3y^2=0$ is twice their product.Preview
  18. Q18Prove that the acute angle $\theta$ between the lines represented by the equation $ax^2+2hxy+by^2=0$ is $\tan\theta=\left|\dfrac{2\sqrt{h^2-…Preview
  19. Q19Find the co-ordinates of the points of intersection of the lines represented by $x^2-y^2-2x+1=0$.Preview
  20. Q20Prove that homogeneous equation of degree two in $x$ and $y$, $ax^2+2hxy+by^2=0$ represents a pair of lines passing through the origin if $h…Preview
  21. Q21Find $k$, if the sum of the slopes of the lines represented by $x^2+kxy-3y^2=0$ is twice their product.Preview
  22. Q22Show that every homogeneous equation of degree two in $x$ and $y$ i.e. $ax^2+2hxy+by^2=0$, represents a pair of lines passing through the or…Preview

More questions

67 Q
+Show 14 questions14 questions
  1. Q51If the equation $4x^2 + hxy + y^2 = 0$ represents two coincident lines, then $h =$ _________. A) $\pm 2$ B) $\pm 3$ C) $\pm 4$ D) $\pm 5$Free
  2. Q52If the lines represented by $kx^2 - 3xy + 6y^2 = 0$ are perpendicular to each other then _________. A) $k = 6$ B) $k = -6$ C) $k = 3$ D) $k…Free
  3. Q53Auxiliary equation of $2x^2 + 3xy - 9y^2 = 0$ is _________. A) $2m^2 + 3m - 9 = 0$ B) $9m^2 - 3m - 2 = 0$ C) $2m^2 - 3m + 9 = 0$ D) $-9m^2 -…Free
  4. Q54The difference between the slopes of the lines represented by $3x^2 - 4xy + y^2 = 0$ is _________. A) $2$ B) $1$ C) $3$ D) $4$Preview
  5. Q55If the two lines $ax^2 + 2hxy + by^2 = 0$ make angles $\alpha$ and $\beta$ with X-axis, then $\tan(\alpha+\beta) =$ _____. A) $\dfrac{h}{a+b…Preview
  6. Q56If the slope of one of the two lines $ax^2 + 2hxy + by^2 = 0$ is twice that of the other, then $ab:h^2 =$ ___. A) $1:2$ B) $2:1$ C) $8:9$ D)…Preview
  7. Q57The joint equation of the lines through the origin and perpendicular to the pair of lines $3x^2 + 4xy - 5y^2 = 0$ is _________. A) $5x^2 + 4…Preview
  8. Q58If acute angle between lines $ax^2 + 2hxy + by^2 = 0$ is $\dfrac{\pi}{4}$ then $4h^2 =$ _________. A) $a^2+4ab+b^2$ B) $a^2+6ab+b^2$ C) $(a+…Preview
  9. Q59If the equation $3x^2 - 8xy + qy^2 + 2x + 14y + p = 1$ represents a pair of perpendicular lines then the values of $p$ and $q$ are respectiv…Preview
  10. Q60The area of triangle formed by the lines $x^2 + 4xy + y^2 = 0$ and $x - y - 4 = 0$ is _________. A) $\dfrac{4}{\sqrt3}$ Sq. units B) $\dfrac…Preview
  11. Q61The combined equation of the co-ordinate axes is _________. A) $x+y=0$ B) $xy=k$ C) $xy=0$ D) $x-y=k$Preview
  12. Q62If $h^2 = ab$, then slope of lines $ax^2+2hxy+by^2=0$ are in the ratio _________. A) $1:2$ B) $2:1$ C) $2:3$ D) $1:1$Preview
  13. Q63If slope of one of the lines $ax^2+2hxy+by^2=0$ is $5$ times the slope of the other, then $5h^2 =$ _________. A) $ab$ B) $2ab$ C) $7ab$ D) $…Preview
  14. Q64If distance between lines $(x-2y)^2 + k(x-2y) = 0$ is $3$ units, then $k =$ ______. A) $\pm\sqrt3$ B) $\pm5\sqrt5$ C) $0$ D) $\pm3\sqrt5$Preview
+Show 53 questions53 questions
  1. Q65Find the joint equation of lines: $x - y = 0$ and $x + y = 0$.Free
  2. Q66Find the joint equation of lines: $x + y - 3 = 0$ and $2x + y - 1 = 0$.Free
  3. Q67Find the joint equation of lines: passing through the origin and having slopes $2$ and $3$.Free
  4. Q68Find the joint equation of lines: passing through the origin and having inclinations $60°$ and $120°$.Preview
  5. Q69Find the joint equation of lines: passing through $(1,2)$ and parallel to the co-ordinate axes.Preview
  6. Q70Find the joint equation of lines: passing through $(3,2)$ and parallel to the line $x = 2$ and $y = 3$.Preview
  7. Q71Find the joint equation of lines: passing through $(-1,2)$ and perpendicular to the lines $x + 2y + 3 = 0$ and $3x - 4y - 5 = 0$.Preview
  8. Q72Find the joint equation of lines: passing through the origin and having slopes $1 + \sqrt3$ and $1 - \sqrt3$.Preview
  9. Q73Find the joint equation of lines: which are at a distance of $9$ units from the Y-axis.Preview
  10. Q74Find the joint equation of lines: passing through the point $(3,2)$, one of which is parallel to the line $x - 2y = 2$ and other is perpendi…Preview
  11. Q75Find the joint equation of lines: passing through the origin and perpendicular to the lines $x + 2y = 19$ and $3x + y = 18$.Preview
  12. Q76Show that each of the following equation represents a pair of lines: $x^2 + 2xy - y^2 = 0$.Preview
  13. Q77Show that each of the following equation represents a pair of lines: $4x^2 + 4xy + y^2 = 0$.Preview
  14. Q78Show that each of the following equation represents a pair of lines: $x^2 - y^2 = 0$.Preview
  15. Q79Show that each of the following equation represents a pair of lines: $x^2 + 7xy - 2y^2 = 0$.Preview
  16. Q80Show that each of the following equation represents a pair of lines: $x^2 - 2\sqrt3xy - y^2 = 0$.Preview
  17. Q81Find the separate equations of lines represented by the following equation: $6x^2 - 5xy - 6y^2 = 0$.Preview
  18. Q82Find the separate equations of lines represented by the following equation: $x^2 - 4y^2 = 0$.Preview
  19. Q83Find the separate equations of lines represented by the following equation: $3x^2 - y^2 = 0$.Preview
  20. Q84Find the separate equations of lines represented by the following equation: $2x^2 + 2xy - y^2 = 0$.Preview
  21. Q85Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by: $x^2 + 4xy - 5y^2 = 0$.Preview
  22. Q86Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by: $2x^2 - 3xy - 9y^2 = 0$.Preview
  23. Q87Find the joint equation of the pair of lines through the origin and perpendicular to the lines given by: $x^2 + xy - y^2 = 0$.Preview
  24. Q88Find $k$ if the sum of the slopes of the lines given by $3x^2 + kxy - y^2 = 0$ is zero.Preview
  25. Q89Find $k$ if the sum of slopes of the lines given by $2x^2 + kxy - 3y^2 = 0$ is equal to their product.Preview
  26. Q90Find $k$ if the slope of one of the lines given by $3x^2 - 4xy + ky^2 = 0$ is $1$.Preview
  27. Q91Find $k$ if one of the lines given by $3x^2 - kxy + 5y^2 = 0$ is perpendicular to the $5x + 3y = 0$.Preview
  28. Q92Find $k$ if the slope of one of the lines given by $3x^2 + 4xy + ky^2 = 0$ is three times the other.Preview
  29. Q93Find $k$ if the slopes of lines given by $kx^2 + 5xy + y^2 = 0$ differ by $1$.Preview
  30. Q94Find $k$ if one of the lines given by $6x^2 + kxy + y^2 = 0$ is $2x + y = 0$.Preview
  31. Q95Find the joint equation of the pair of lines which bisect angle between the lines given by $x^2 + 3xy + 2y^2 = 0$.Preview
  32. Q96Find the joint equation of the pair of lines through the origin and making equilateral triangle with the line $x = 3$.Preview
  33. Q97Show that the lines $x^2 - 4xy + y^2 = 0$ and $x + y = 10$ contain the sides of an equilateral triangle. Find the area of the triangle.Preview
  34. Q98If the slope of one of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is three times the other then prove that $3h^2 = 4ab$.Preview
  35. Q99Find the combined equation of the bisectors of the angles between the lines represented by $5x^2 + 6xy - y^2 = 0$.Preview
  36. Q100Find $a$ if the sum of slope of lines represented by $ax^2 + 8xy + 5y^2 = 0$ is twice their product.Preview
  37. Q101If line $4x - 5y = 0$ coincides with one of the lines given by $ax^2 + 2hxy + by^2 = 0$ then show that $25a + 40h + 16b = 0$.Preview
  38. Q102Show that the following equations represent a pair of lines, find the acute angle between them: $9x^2 - 6xy + y^2 + 18x - 6y + 8 = 0$.Preview
  39. Q103Show that the following equations represent a pair of lines, find the acute angle between them: $2x^2 + xy - y^2 + x + 4y - 3 = 0$.Preview
  40. Q104Show that the following equations represent a pair of lines, find the acute angle between them: $(x-3)^2 + (x-3)(y-4) - 2(y-4)^2 = 0$.Preview
  41. Q105Find the combined equation of pair of lines through the origin each of which makes angle of $60°$ with the Y-axis.Preview
  42. Q106If lines represented by $ax^2 + 2hxy + by^2 = 0$ make angles of equal measures with the co-ordinate axes then show that $\dfrac{a}{b} = \pm1…Preview
  43. Q107Show that the combined equation of a pair of lines through the origin and each making an angle of $\alpha$ with the line $x + y = 0$ is $x^2…Preview
  44. Q108Show that the line $3x + 4y + 5 = 0$ and the lines $(3x+4y)^2 - 3(4x-3y)^2 = 0$ form an equilateral triangle.Preview
  45. Q109Show that lines $x^2 - 4xy + y^2 = 0$ and $x + y = 6$ form an equilateral triangle. Find its area and perimeter.Preview
  46. Q110If the slope of one of the lines given by $ax^2 + 2hxy + by^2 = 0$ is square of the other then show that $a^2b + ab^2 + 8h^3 = 6abh$.Preview
  47. Q111Prove that the product of lengths of perpendiculars drawn from $P(x_1, y_1)$ to the lines represented by $ax^2 + 2hxy + by^2 = 0$ is $\dfrac…Preview
  48. Q112Show that the difference between the slopes of lines given by $(\tan^2\theta + \cos^2\theta)x^2 - 2xy\tan\theta + (\sin^2\theta)y^2 = 0$ is…Preview
  49. Q113Find the condition that the equation $ay^2 + bxy + ex + dy = 0$ may represent a pair of lines.Preview
  50. Q114If the lines given by $ax^2 + 2hxy + by^2 = 0$ form an equilateral triangle with the line $lx + my = 1$ then show that $(3a+b)(a+3b) = 4h^2$…Preview
  51. Q115If line $x + 2 = 0$ coincides with one of the lines represented by the equation $x^2 + 2xy + 4y + k = 0$ then show that $k = -4$.Preview
  52. Q116Prove that the combined equation of the pair of lines passing through the origin and perpendicular to the lines represented by $ax^2 + 2hxy…Preview
  53. Q117If equation $ax^2 - y^2 + 2y + c = 1$ represents a pair of perpendicular lines then find $a$ and $c$.Preview