Straight Line — Class 11 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Straight Line, written for Class 11 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
Straight Line in one look
a locus is a set of points obeying a rule, and a line is the simplest set whose slope never changes
Locus and the Equation of a Locus
- a locus is the set of all points satisfying one geometric condition, plural loci
- an equation OF a locus holds on every point of it, none off it
- method: let P(x,y) be general, translate the condition into an x,y equation, simplify
- e.g. perpendicular bisector of AB: locus of P with PA=PB, a line
- e.g. bisector of an angle: locus of points equidistant from its arms, a ray
Q. Find the equation of the locus of points equidistant from A(-3,0) and B(3,0). Identify the locus.
- let P(x,y) satisfy PA=PB, so PA^2=PB^2
- (x+3)^2+y^2=(x-3)^2+y^2
- x^2+6x+9=x^2-6x+9, so 12x=0
Answer: x=0: the locus is the Y-axis, the perpendicular bisector of AB
Shift of Origin
- shift the origin to O'(h,k), axes kept parallel: x=x'+h, y=y'+k
- to move a locus into the new axes, substitute x'+h for x, y'+k for y, simplify
Q. The origin is shifted to (2,3), axes parallel. Find the new equation of x^2-xy-2y^2-x+4y+2=0
- substitute x=X+2, y=Y+3
- (X+2)^2-(X+2)(Y+3)-2(Y+3)^2-(X+2)+4(Y+3)+2=0
- expand and collect every constant and cross term
Answer: X^2-XY-2Y^2-10Y-8=0
Inclination and Slope
- inclination θ: smallest angle a line makes with the positive X-axis, anticlockwise; 0°≤θ<180°
- slope m=tanθ when it exists; undefined for the Y-axis and lines parallel to it
- slope of the X-axis, and of lines parallel to it, is 0
- parallel lines share the same inclination, equivalently the same slope
- two points A(x_1,y_1), B(x_2,y_2) on a non-vertical line: m=y_2-y_1/x_2-x_1
Perpendicular and Angle Between Lines
- perpendicular
m_1m_2=-1 - acute angle between two lines
tanθ=left|m_1-m_2/1+m_1m_2right|
Q. Find the acute angle between lines of slope 3 and -2
- tanθ=left|3-(-2)/1+(3)(-2)right|=left|5/-5right|=1
Answer: tanθ=1, so θ=45°
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Straight Line sheet covers
- Locus and the Equation of a Locus
- Shift of Origin
- Inclination and Slope
- Perpendicular and Angle Between Lines
- perpendicular
- acute angle between two lines
- The Five Standard Forms
- Two-point form
- Normal form
- General Form
- Distance Formulas
- origin to line ax+by+c=0
- point (x_1,y_1) to line ax+by+c=0
- between parallels ax+by+c_1=0, ax+by+c_2=0
- Family of Lines: u+kv=0
- Point of Intersection, and What a Line's Sign Tells You
- Parallel to the X-axis
- Angle fixes the other slope
- 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 Straight Line 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.