JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
STRAIGHT LINES
— JEE MATH APEX —
JEE MATH APEX
1. Distance Formula
📌 Distance Between Two Points
• Distance between P(x₁,y₁) and Q(x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²]
• Distance from origin to P(x,y): OP = √(x² + y²)
• Distance from origin to P(x,y): OP = √(x² + y²)
JEE MATH APEX
2. Section Formula
📌 Internal, External & Midpoint
• Internal division (m:n): x = (mx₂ + nx₁)/(m+n), y = (my₂ + ny₁)/(m+n)
• External division (m:n): x = (mx₂ − nx₁)/(m−n), y = (my₂ − ny₁)/(m−n)
• Midpoint: x = (x₁+x₂)/2, y = (y₁+y₂)/2
• Centroid of triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
• Incentre: I = ((ax₁+bx₂+cx₃)/(a+b+c), (ay₁+by₂+cy₃)/(a+b+c))
• External division (m:n): x = (mx₂ − nx₁)/(m−n), y = (my₂ − ny₁)/(m−n)
• Midpoint: x = (x₁+x₂)/2, y = (y₁+y₂)/2
• Centroid of triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
• Incentre: I = ((ax₁+bx₂+cx₃)/(a+b+c), (ay₁+by₂+cy₃)/(a+b+c))
JEE MATH APEX
3. Area of Triangle
📌 Area Using Coordinates
• Area of triangle: Δ = (1/2)|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|
• Collinearity condition: x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) = 0
• Area of quadrilateral: (1/2)|x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ − (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)|
• Collinearity condition: x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂) = 0
• Area of quadrilateral: (1/2)|x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁ − (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁)|
JEE MATH APEX
4. Slope of a Line
📌 Slope Formula
• Slope of line through P(x₁,y₁) and Q(x₂,y₂): m = (y₂ − y₁)/(x₂ − x₁)
• Slope of line with inclination θ: m = tan θ
• Slope of horizontal line: m = 0
• Slope of vertical line: m = ∞ (undefined)
• Slope of line with inclination θ: m = tan θ
• Slope of horizontal line: m = 0
• Slope of vertical line: m = ∞ (undefined)
JEE MATH APEX
5. Forms of Straight Line
📌 Various Forms of Line Equation
• Slope-intercept form: y = mx + c
• Point-slope form: y − y₁ = m(x − x₁)
• Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)
• Intercept form: x/a + y/b = 1
• Normal form: x cos α + y sin α = p
• Point-slope form: y − y₁ = m(x − x₁)
• Two-point form: (y − y₁)/(y₂ − y₁) = (x − x₁)/(x₂ − x₁)
• Intercept form: x/a + y/b = 1
• Normal form: x cos α + y sin α = p
JEE MATH APEX
6. General Equation & Slope
📌 Ax + By + C = 0
• General form: Ax + By + C = 0
• Slope: m = −A/B
• x-intercept: −C/A
• y-intercept: −C/B
• Convert to slope-intercept: y = (−A/B)x − C/B
• Slope: m = −A/B
• x-intercept: −C/A
• y-intercept: −C/B
• Convert to slope-intercept: y = (−A/B)x − C/B
JEE MATH APEX
7. Angle Between Two Lines
📌 Angle Formula
• Angle between two lines with slopes m₁ and m₂: tan θ = |(m₁ − m₂)/(1 + m₁m₂)|
• Parallel condition: m₁ = m₂
• Perpendicular condition: m₁ × m₂ = −1
• For lines A₁x+B₁y+C₁=0 and A₂x+B₂y+C₂=0: tan θ = |(A₁B₂ − A₂B₁)/(A₁A₂ + B₁B₂)|
• Parallel condition: m₁ = m₂
• Perpendicular condition: m₁ × m₂ = −1
• For lines A₁x+B₁y+C₁=0 and A₂x+B₂y+C₂=0: tan θ = |(A₁B₂ − A₂B₁)/(A₁A₂ + B₁B₂)|
JEE MATH APEX
8. Distance of a Point from Line
📌 Perpendicular Distance
• Distance of point (x₁,y₁) from line Ax+By+C=0: d = |Ax₁ + By₁ + C| / √(A² + B²)
• Distance from origin: d = |C| / √(A² + B²)
• Distance between two parallel lines Ax+By+C₁=0 and Ax+By+C₂=0: d = |C₁ − C₂| / √(A² + B²)
• Distance from origin: d = |C| / √(A² + B²)
• Distance between two parallel lines Ax+By+C₁=0 and Ax+By+C₂=0: d = |C₁ − C₂| / √(A² + B²)
JEE MATH APEX
9. Foot of Perpendicular
📌 Foot & Image of a Point
• Foot of perpendicular from (x₁,y₁) to Ax+By+C=0:
(x−x₁)/A = (y−y₁)/B = −(Ax₁+By₁+C)/(A²+B²)
• Image of point: If foot is (h,k), image is (2h−x₁, 2k−y₁)
• Reflection of line: Use foot of perpendicular
(x−x₁)/A = (y−y₁)/B = −(Ax₁+By₁+C)/(A²+B²)
• Image of point: If foot is (h,k), image is (2h−x₁, 2k−y₁)
• Reflection of line: Use foot of perpendicular
JEE MATH APEX
10. Concurrent Lines
📌 Condition for Concurrency
• Three lines A₁x+B₁y+C₁=0, A₂x+B₂y+C₂=0, A₃x+B₃y+C₃=0 are concurrent if:
|A₁ B₁ C₁; A₂ B₂ C₂; A₃ B₃ C₃| = 0 (determinant)
• Or: A₁(B₂C₃−B₃C₂) + B₁(C₂A₃−C₃A₂) + C₁(A₂B₃−A₃B₂) = 0
|A₁ B₁ C₁; A₂ B₂ C₂; A₃ B₃ C₃| = 0 (determinant)
• Or: A₁(B₂C₃−B₃C₂) + B₁(C₂A₃−C₃A₂) + C₁(A₂B₃−A₃B₂) = 0
JEE MATH APEX
11. Family of Lines
📌 Lines Through Intersection
• Family of lines through intersection of L₁=0 and L₂=0:
L₁ + λL₂ = 0 (where λ is parameter)
• Family of lines parallel to Ax+By+C=0: Ax + By + λ = 0
• Family of lines perpendicular to Ax+By+C=0: Bx − Ay + λ = 0
L₁ + λL₂ = 0 (where λ is parameter)
• Family of lines parallel to Ax+By+C=0: Ax + By + λ = 0
• Family of lines perpendicular to Ax+By+C=0: Bx − Ay + λ = 0
JEE MATH APEX
12. Angle Bisectors
📌 Equations of Angle Bisectors
• Angle bisectors of lines A₁x+B₁y+C₁=0 and A₂x+B₂y+C₂=0:
(A₁x+B₁y+C₁)/√(A₁²+B₁²) = ± (A₂x+B₂y+C₂)/√(A₂²+B₂²)
• Acute angle bisector: Use the sign that makes angle acute
• Obtuse angle bisector: Other sign
(A₁x+B₁y+C₁)/√(A₁²+B₁²) = ± (A₂x+B₂y+C₂)/√(A₂²+B₂²)
• Acute angle bisector: Use the sign that makes angle acute
• Obtuse angle bisector: Other sign
JEE MATH APEX
13. Pair of Straight Lines
📌 Joint Equation
• General joint equation: ax² + 2hxy + by² + 2gx + 2fy + c = 0
• Represents pair of lines if: abc + 2fgh − af² − bg² − ch² = 0
• Angle between pair: tan θ = 2√(h²−ab)/(a+b)
• Lines are parallel if: h² = ab
• Lines are perpendicular if: a + b = 0
• Represents pair of lines if: abc + 2fgh − af² − bg² − ch² = 0
• Angle between pair: tan θ = 2√(h²−ab)/(a+b)
• Lines are parallel if: h² = ab
• Lines are perpendicular if: a + b = 0
JEE MATH APEX
14. Shifting of Origin
📌 Translation of Axes
• If origin shifts to (h,k): x = X + h, y = Y + k
• New coordinates: X = x − h, Y = y − k
• Line Ax+By+C=0 becomes: AX + BY + (Ah+Bk+C) = 0
• New coordinates: X = x − h, Y = y − k
• Line Ax+By+C=0 becomes: AX + BY + (Ah+Bk+C) = 0
JEE MATH APEX
15. Rotation of Axes
📌 Rotation of Coordinate Axes
• If axes rotated by angle θ:
x = X cos θ − Y sin θ
y = X sin θ + Y cos θ
• Inverse: X = x cos θ + y sin θ, Y = −x sin θ + y cos θ
x = X cos θ − Y sin θ
y = X sin θ + Y cos θ
• Inverse: X = x cos θ + y sin θ, Y = −x sin θ + y cos θ
✅ STRAIGHT LINES ✅
— JEE MATH APEX —
— JEE MATH APEX —
JEE MATH APEX 
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