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Mathematics · Class 11 Science

Ch 17Pair of Straight Lines — Class 11 Mathematics, concept-first.

Two straight lines through the origin can always be written as and , since any line through the origin has an equation with zero constant term. Multiplying the two equations together gives a single equation satisfied by every point that lies on either line:

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Key concepts

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Combined Equation of a Pair of Lines Through the Origin

Whenever two straight lines pass through the origin, each has an equation of the form (no constant term, since must satisfy it). If the two lines are and , multiplying them together produces a single equation,

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

4.1

Combined Equation of a Pair of Lines Through the Origin

Two straight lines through the origin can always be written as and , since any line through the origin has an equation with zero constant term.

4.2

Condition for Real and Distinct Lines ($h^2>ab$)

Given a homogeneous second-degree equation , we must ask: does this always represent two real straight lines through the origin? To answer this, treat the equation as a quadratic in the ratio (assumin…

4.3

Angle Between a Pair of Lines

Suppose represents two real lines through the origin (so , as in the previous section), with slopes and .

4.4

Perpendicularity and Coincidence Conditions

The angle formula derived in the previous section, , has two important special cases that are asked about far more often than the general angle itself: when the lines are perpendicular () and when the…

+Exercise 4(a)i9 questions
  1. Q1Find the equations of the pair of lines represented by $2x^2 - 3xy - 2y^2 = 0$.Free
  2. Q2Find the equations of the pair of lines represented by $x^2 + xy - 6y^2 = 0$.Free
  3. Q3Find the equations of the pair of lines represented by $3x^2 + 7xy + 2y^2 = 0$.Free
  4. Q4Find the angle between the pair of lines represented by $x^2 - 4xy + y^2 = 0$.Preview
  5. Q5Find the angle between the pair of lines represented by $3x^2 + 10xy + 3y^2 = 0$.Preview
  6. Q6Find the angle between the pair of lines represented by $2x^2 - 7xy + 3y^2 = 0$.Preview
  7. Q7Find the value of $\lambda$ for which $\lambda x^2 + 8xy + 2y^2 = 0$ represents a pair of perpendicular lines.Preview
  8. Q8Find the value of $\lambda$ for which $x^2 + \lambda xy + 9y^2 = 0$ represents a pair of coincident lines.Preview
  9. Q9Find the value of $m$ for which $mx^2 - 10xy + 8y^2 = 0$ represents a pair of coincident lines.Preview
4.5

Angle-Bisector Pair of a Homogeneous Pair of Lines

Given a pair of lines through the origin, the two lines that bisect the angles between them are also two straight lines through the origin, and they too can be written as one combined equation.

4.6

General Second-Degree Equation Representing a Pair of Lines

So far every pair of lines considered has passed through the origin, giving a purely homogeneous equation.

4.7

Distance Between Parallel Lines and Point of Intersection

Once we know that represents a genuine pair of straight lines (via the condition of the previous section), two further practical questions arise: if the two lines are parallel, how far apart are they;…

+Exercise 4(b)i7 questions
  1. Q1Find the combined equation of the pair of bisectors of the angles between the lines represented by $x^2 - 3xy + 2y^2 = 0$.Free
  2. Q2Find the combined equation of the pair of bisectors of the angles between the lines represented by $2x^2 + 5xy + 2y^2 = 0$.Free
  3. Q3Find the combined equation of the pair of bisectors of the angles between the lines represented by $3x^2 - 5xy - 2y^2 = 0$.Free
  4. Q4Show that $2x^2 + 3xy - 2y^2 + x + 7y - 3 = 0$ represents a pair of straight lines, and find the separate equations of the two lines.Preview
  5. Q5Find the point of intersection of the pair of lines represented by $x^2 - y^2 - x + 5y - 6 = 0$.Preview
  6. Q6Find the distance between the pair of parallel lines represented by $4x^2 + 12xy + 9y^2 - 10x - 15y + 4 = 0$.Preview
  7. Q7Find the values of $k$ for which $x^2 + kxy + y^2 - 5x - 7y + 6 = 0$ represents a pair of straight lines.Preview
4.8

Homogenising Technique for Lines Joining the Origin to Curve Intersections

A recurring type of problem asks: given a curve (typically a conic such as a circle, ellipse, or parabola) and a line that cuts it at two points and , find the combined equation of the two lines and j…

+Exercise 4(c)i5 questions
  1. Q1Find the combined equation of the lines joining the origin to the points of intersection of the circle $x^2 + y^2 = 9$ and the line $2x + y…Free
  2. Q2Find the combined equation of the lines joining the origin to the points of intersection of the ellipse $9x^2 + 4y^2 = 36$ and the line $x +…Free
  3. Q3Find the value of $k$ for which the lines joining the origin to the points of intersection of the curve $x^2 + y^2 - 2x - 2y + 1 = 0$ and th…Preview
  4. Q4Find the combined equation of the lines joining the origin to the points where the line $x - 2y + 4 = 0$ meets the parabola $y^2 = 4x$, and…Preview
  5. Q5Find the combined equation of the lines joining the origin to the points of intersection of the circle $x^2 + y^2 = 8$ and the line $x - y =…Preview

Sample & Board Papers

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

+Show 14 questions14 questions
  1. Q1Suppose that $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines. If $\theta$ is the angle between them, show that $\cos\theta = \d…Preview
  2. Q2Show that the lines joining the origin to the points of intersection of the straight line $x - y - \sqrt{2} = 0$ and the curve $x^2 - xy + y…Preview
  3. Q3Show that the area of the triangle formed by the lines $ax^{2} + 2hxy + by^{2} = 0$ and $lx + my + n = 0$ is $\left|\dfrac{n^{2}\sqrt{h^{2}…Preview
  4. Q4Find the values of $k$, if the lines joining the origin to the points of intersection of the curve $2x^{2} - 2xy + 3y^{2} + 2x - y + 1 = 0$…Preview
  5. Q5If the equation $S \equiv ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ represents a pair of parallel straight lines, then show that (i) $h^2 = ab…Preview
  6. Q6Show that the lines joining the origin to the points of intersection of the curve $x^2 - xy + y^2 + 3x + 3y - 2 = 0$ and the straight line $…Preview
  7. Q7Show that the lines represented by $(lx+my)^2 - 3(mx-ly)^2 = 0$ and $lx + my + n = 0$ form an equilateral triangle with area $\frac{n^2}{\sq…Preview
  8. Q8Find the condition for the chord $lx + my = 1$ of the circle $x^2 + y^2 = a^2$ (whose centre is the origin) to subtend a right angle at the…Preview
  9. Q9The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of straight lines and $\theta$ is the angle between the lines. Then show that $$\cos…Preview
  10. Q10Show that the lines joining the origin to the points of intersection of the curve $x^2 - xy + y^2 + 3x + 3y - 2 = 0$ and the straight line $…Preview
  11. Q11If the equation $ax^2 + 2hxy + by^2 = 0$ represent a pair of straight lines and $\theta$ is the angle between the lines then prove that $\co…Preview
  12. Q12Find the values of $k$, if the lines joining the origin to the points of intersection of the curve $2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0$ and…Preview
  13. Q13Show that the area of the triangle formed by the lines $ax^2 + 2hxy + by^2 = 0$ and $lx + my + n = 0$ is $\left| \dfrac{n^2 \sqrt{h^2 - ab}}…Preview
  14. Q14Find the values of $k$, if the lines joining the origin to the points of intersection of the curve $2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0$ and…Preview