Pair of Straight Lines — Class 12 Maharashtra HSC Mathematics revision notes
A 6-page visual revision sheet for Pair of Straight Lines, written for Class 12 Maharashtra HSC Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Pair of Straight Lines in one look
one product equation hides two lines, and h squared minus ab says everything about how they sit
Combined Equation of a Pair of Lines
- let u=a_1x+b_1y+c_1, v=a_2x+b_2y+c_2; each of u=0, v=0 is an ordinary line
- uv=0 is the combined equation of u=0 and v=0, true exactly on either line
- u=0, v=0 are the separate equations; factorising the combined equation recovers them
- one product, two lines: uv=0 stands for BOTH lines, never their intersection point alone
Combined Equation, worked
- the combined equation of u=0 and v=0 is uv=0, so multiply the two expressions directly
- (x+y-2)(2x-y+2)=0
- expand: x(2x-y+2)+y(2x-y+2)-2(2x-y+2)
- =2x^2-xy+2x+2xy-y^2+2y-4x+2y-4
- collect like terms: 2x^2+xy-y^2-2x+4y-4
- **2x^2+xy-y^2-2x+4y-4=0** is the combined equation
Separate equations from a combined equation
Q. find the separate equations of the lines represented by 6x^2-5xy-6y^2=0
- split the middle term so each pair shares a common factor: 6x^2-9xy+4xy-6y^2=0
- group: 3x(2x-3y)+2y(2x-3y)=0
- factor out the common bracket: (2x-3y)(3x+2y)=0
Answer: the separate equations are 2x-3y=0 and 3x+2y=0
Page 1 of 6. The full sheet, all 6 pages, opens free in the app.
What this Pair of Straight Lines sheet covers
- Combined Equation of a Pair of Lines
- Combined Equation, worked
- Separate equations from a combined equation
- Homogeneous Equation of Degree Two
- The Test: when does the homogeneous pair give real lines?
- represents a pair of REAL lines through the origin exactly when
- the two lines are distinct when
- the two lines are coincident (the same line, twice over) when
- no real pair of lines exists when
- Special Cases of the Homogeneous Pair
- Slopes of the Pair, and the Auxiliary Equation
- sum of the slopes (needs b≠0)
- product of the slopes
- m_1,m_2 are the roots of the auxiliary equation
- Separate equations by the auxiliary equation, worked
- Angle Between the Lines of a Homogeneous Pair
- the acute angle θ between the lines represented by ax^2+2hxy+by^2=0
- the lines are perpendicular exactly when
- the lines are coincident exactly when
- Acute angle between a homogeneous pair
- Bisectors of the Angle Between the Pair
- the combined equation of the two angle bisectors of ax^2+2hxy+by^2=0
- General Second Degree Equation
- Necessary Conditions for a Pair of Lines
- condition one, the 3×3 determinant vanishes
- condition two, same test as the homogeneous case
- Homogeneous Pair vs General Second-Degree Pair
- Homogeneous, ax²+2hxy+by²=0
- General, plus 2gx+2fy+c
- Once It IS a Pair of Lines
- perpendicular exactly when
- parallel (or coincident) exactly when
- point of intersection, valid when h^2-ab≠0
- A general second degree equation, fully worked
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Pair of Straight Lines the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.