MH HSC 12 Maths

Pair of Straight Lines

Each concept is a short lesson, then questions until you get three right.

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9 concepts, a short lesson each

Lines through the origin0/7
General equation0/2

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Combined equation of two lines

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

Lines through the origin0 of 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 twoStart
Separate equations from the combined oneFactorise $ax^2 + 2hxy + by^2 = 0$ , or solve $bm^2 + 2hm + a = 0$ for the slopes $m = y/x$ Start
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 slopesStart
Angle between the lines $\tan\theta = \left|\dfrac{2\sqrt{h^2 - ab}}{a + b}\right|$ ; perpendicular when $a + b = 0$ , coincident when $h^2 = ab$ Start
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 $\tan\alpha$ Start
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 equilateralStart
Bisectors of the angles between the linesThe joint equation of the bisectors of $ax^2 + 2hxy + by^2 = 0$ is $h(x^2 - y^2) = (a - b)xy$ Start

General equation0 of 2

Practise Pair of Straight Lines

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answers Past board questionsAsked from this chapter, with solutions