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
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
JEE MATH APEX
ELLIPSE
— JEE MATH APEX —
JEE MATH APEX
1. Standard Equation of Ellipse
📌 Standard Forms
• Horizontal ellipse: x²/a² + y²/b² = 1 (a > b)
• Centre: (0, 0)
• Major axis: x-axis (length 2a)
• Minor axis: y-axis (length 2b)
• Vertical ellipse: x²/b² + y²/a² = 1 (a > b)
• Vertices: (±a, 0), Foci: (±ae, 0)
• Centre: (0, 0)
• Major axis: x-axis (length 2a)
• Minor axis: y-axis (length 2b)
• Vertical ellipse: x²/b² + y²/a² = 1 (a > b)
• Vertices: (±a, 0), Foci: (±ae, 0)
JEE MATH APEX
2. Key Parameters
📌 Eccentricity, Foci, Directrix
• Eccentricity: e = √(1 − b²/a²) (0 < e < 1)
• b² = a²(1 − e²)
• Foci: (±ae, 0)
• Directrices: x = ±a/e
• Latus rectum: 2b²/a
• Ends of latus rectum: (±ae, ±b²/a)
• b² = a²(1 − e²)
• Foci: (±ae, 0)
• Directrices: x = ±a/e
• Latus rectum: 2b²/a
• Ends of latus rectum: (±ae, ±b²/a)
JEE MATH APEX
3. Parametric Form
📌 Parametric Coordinates
• For x²/a² + y²/b² = 1: x = a cos θ, y = b sin θ
• Parametric point: (a cos θ, b sin θ)
• θ is called eccentric angle
• Auxiliary circle: x² + y² = a²
• Eccentric angle of any point on ellipse is angle of corresponding point on auxiliary circle
• Parametric point: (a cos θ, b sin θ)
• θ is called eccentric angle
• Auxiliary circle: x² + y² = a²
• Eccentric angle of any point on ellipse is angle of corresponding point on auxiliary circle
JEE MATH APEX
4. Tangent to Ellipse
📌 Equation of Tangent
• At point (x₁,y₁): xx₁/a² + yy₁/b² = 1
• At parametric point (a cos θ, b sin θ): (x cos θ)/a + (y sin θ)/b = 1
• With slope m: y = mx ± √(a²m² + b²)
• Point of contact: (±a²m/√(a²m²+b²), ∓b²/√(a²m²+b²))
• Condition for tangency: c = ±√(a²m² + b²)
• At parametric point (a cos θ, b sin θ): (x cos θ)/a + (y sin θ)/b = 1
• With slope m: y = mx ± √(a²m² + b²)
• Point of contact: (±a²m/√(a²m²+b²), ∓b²/√(a²m²+b²))
• Condition for tangency: c = ±√(a²m² + b²)
JEE MATH APEX
5. Normal to Ellipse
📌 Equation of Normal
• At point (x₁,y₁): a²x/x₁ − b²y/y₁ = a² − b²
• At parametric point (a cos θ, b sin θ): ax sec θ − by cosec θ = a² − b²
• With slope m: y = mx ± m(a²−b²)/√(a²+b²m²)
• Four normals can be drawn from a point to ellipse
• At parametric point (a cos θ, b sin θ): ax sec θ − by cosec θ = a² − b²
• With slope m: y = mx ± m(a²−b²)/√(a²+b²m²)
• Four normals can be drawn from a point to ellipse
JEE MATH APEX
6. Chord of Contact
📌 Chord of Contact & Pair of Tangents
• Chord of contact from (x₁,y₁): xx₁/a² + yy₁/b² = 1
• Pair of tangents: SS₁ = T²
• Where S = x²/a² + y²/b² − 1, S₁ = x₁²/a² + y₁²/b² − 1, T = xx₁/a² + yy₁/b² − 1
• Pair of tangents: SS₁ = T²
• Where S = x²/a² + y²/b² − 1, S₁ = x₁²/a² + y₁²/b² − 1, T = xx₁/a² + yy₁/b² − 1
JEE MATH APEX
7. Chord with Midpoint
📌 Equation of Chord with Given Midpoint
• Chord with midpoint (x₁,y₁): T = S₁
• xx₁/a² + yy₁/b² − 1 = x₁²/a² + y₁²/b² − 1
• Slope of chord with midpoint (x₁,y₁): −b²x₁/(a²y₁)
• xx₁/a² + yy₁/b² − 1 = x₁²/a² + y₁²/b² − 1
• Slope of chord with midpoint (x₁,y₁): −b²x₁/(a²y₁)
JEE MATH APEX
8. Director Circle
📌 Director Circle of Ellipse
• Director circle of x²/a² + y²/b² = 1: x² + y² = a² + b²
• Locus of intersection of perpendicular tangents
• Radius of director circle: √(a² + b²)
• Locus of intersection of perpendicular tangents
• Radius of director circle: √(a² + b²)
JEE MATH APEX
9. Focal Distances
📌 Sum of Focal Distances
• For any point P on ellipse: SP + S’P = 2a (constant)
• Focal distance of point (x₁,y₁): SP = a − ex₁ and S’P = a + ex₁
• Sum of focal distances = Major axis length = 2a
• Distance from focus to directrix = a/e
• Focal distance of point (x₁,y₁): SP = a − ex₁ and S’P = a + ex₁
• Sum of focal distances = Major axis length = 2a
• Distance from focus to directrix = a/e
JEE MATH APEX
10. Position of a Point
📌 Point Relative to Ellipse
• For ellipse x²/a² + y²/b² = 1 and point (x₁,y₁):
• If x₁²/a² + y₁²/b² < 1: Point inside
• If x₁²/a² + y₁²/b² = 1: Point on ellipse
• If x₁²/a² + y₁²/b² > 1: Point outside
• If x₁²/a² + y₁²/b² < 1: Point inside
• If x₁²/a² + y₁²/b² = 1: Point on ellipse
• If x₁²/a² + y₁²/b² > 1: Point outside
JEE MATH APEX
11. Chord Joining Two Points
📌 Chord Between Two Parametric Points
• Chord joining θ₁ and θ₂:
(x/a)cos[(θ₁+θ₂)/2] + (y/b)sin[(θ₁+θ₂)/2] = cos[(θ₁−θ₂)/2]
• If θ₁+θ₂ = constant, chord passes through fixed point
• Slope of chord: −b/a cot[(θ₁+θ₂)/2]
(x/a)cos[(θ₁+θ₂)/2] + (y/b)sin[(θ₁+θ₂)/2] = cos[(θ₁−θ₂)/2]
• If θ₁+θ₂ = constant, chord passes through fixed point
• Slope of chord: −b/a cot[(θ₁+θ₂)/2]
JEE MATH APEX
12. Reflection Property
📌 Reflection Property of Ellipse
• Tangent at any point makes equal angles with focal radii
• Ray from one focus reflects to other focus
• Normal bisects the angle between focal radii
• Tangent bisects the exterior angle between focal radii
• Ray from one focus reflects to other focus
• Normal bisects the angle between focal radii
• Tangent bisects the exterior angle between focal radii
JEE MATH APEX
13. Area of Ellipse
📌 Area & Circumference
• Area of ellipse: πab
• Circumference (approximate): π[3(a+b) − √((3a+b)(a+3b))]
• Area of quadrilateral formed by tangents at parametric points θ₁, θ₂, θ₃, θ₄: 4ab/tan[(θ₁−θ₂)/2]
• Circumference (approximate): π[3(a+b) − √((3a+b)(a+3b))]
• Area of quadrilateral formed by tangents at parametric points θ₁, θ₂, θ₃, θ₄: 4ab/tan[(θ₁−θ₂)/2]
JEE MATH APEX
14. Important Theorems (Advanced)
📌 Key Results for JEE Advanced
• If normal at θ₁ meets ellipse again at θ₂: tan(θ₁/2)·tan(θ₂/2) = −(a−b)/(a+b)
• Four normals pass through a point if it lies inside the evolute
• Evolute of ellipse: (ax)^(2/3) + (by)^(2/3) = (a²−b²)^(2/3)
• Maximum area of triangle inscribed in ellipse: (3√3/4)ab
• Four normals pass through a point if it lies inside the evolute
• Evolute of ellipse: (ax)^(2/3) + (by)^(2/3) = (a²−b²)^(2/3)
• Maximum area of triangle inscribed in ellipse: (3√3/4)ab
✅ ELLIPSE ✅
— JEE MATH APEX —
— JEE MATH APEX —
JEE MATH APEX 
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