MH Board · Chapter

Pair of Straight Lines

The concepts MH Board needs, one connected idea at a time.

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Maharashtra Board

MH Board · Chapter

Pair of Straight Lines

The concepts MH Board needs, one connected idea at a time.

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concepts mastered

Lines through the origin0/7
General equation0/2

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Up nextCombined equation of two linesMultiply the two equations, each written $= 0$ : $(a_1x + b_1y)(a_2x + b_2y) = 0$ gives a homogeneous equation of degree twoStart

Whole chapter

How the ideas build on each other. Each one opens once you've mastered what it needs.

Lines through the origin0 / 7

Combined equation of two linesMultiply the two equations, each written $= 0$ : $(a_1x + b_1y)(a_2x + b_2y) = 0$

gives a homogeneous equation of degree two

Start

Separate equations from the combined oneFactorise

ax^2 + 2hxy + by^2 = 0

, or solve

bm^2 + 2hm + a = 0

for the slopes

m = y/x

Locked

Sum and product of slopes

m_1 + m_2 = -\dfrac{2h}{b}

m_1m_2 = \dfrac{a}{b}

; use them to find a missing coefficient from a condition on the slopesLocked

Angle between the lines

\tan\theta = \left|\dfrac{2\sqrt{h^2 - ab}}{a + b}\right|

; perpendicular when

a + b = 0

, coincident when

h^2 = ab

Locked

Pairs perpendicular or at an angle to given linesThrough the origin, perpendicular to the pair

ax^2 + 2hxy + by^2 = 0

bx^2 - 2hxy + ay^2 = 0

; at an angle

\alpha

to a line, from

\tan\alpha

Locked

Triangles formed with a third lineFind where the pair meets the third line, then the angles or the area of the triangle; equal angles of 60° make it equilateralLocked

Bisectors of the angles between the linesThe joint equation of the bisectors of

ax^2 + 2hxy + by^2 = 0

h(x^2 - y^2) = (a - b)xy

Locked

General equation0 / 2

When a general equation is a pair of lines $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is a pair of lines when $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix} = 0$ and $h^2 \ge ab$ Locked
Angle, parallel pair and point of intersectionThe angle depends only on $a, h, b$ ; the pair through a point parallel to $ax^2 + 2hxy + by^2 = 0$

shifts

x

and

y

;

the lines meet where the two separated lines cross

Locked