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JEE Main & Advanced Mathematics Conic Sections Intersection of a Line and an Ellipse

Intersection of a Line and an Ellipse

Category : JEE Main & Advanced

The line \[y=mx+c\] intersects the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] in two distinct points if \[{{a}^{2}}{{m}^{2}}+{{b}^{2}}>{{c}^{2}}\], in one point if \[{{c}^{2}}={{a}^{2}}{{m}^{2}}+{{b}^{2}}\] and does not intersect if \[{{a}^{2}}{{m}^{2}}+{{b}^{2}}<{{c}^{2}}\].

Position of a Point with Respect to an Ellipse

Equations of Tangent in Different Forms

Other Topics

Definition
Definitions of Various Important Terms
General Equation of a Conic Section When Its Focus, Directrix and Eccentricity are Given
Recognisation of Conics
Definition
Standard Equation of The Parabola
Special form of Parabola \[{{\mathbf{(yk)}}^{\mathbf{2~}}}\mathbf{= 4a}\,\,\mathbf{(xh)= a}\]
Parametric Equations of a Parabola
Position of a Point and a Line With Respect to a Parabola
Equations of Tangent in Different Forms
Point of Intersection of Tangents at Any Two Points on The Parabola
Equation of Pair of Tangents From a Point to a Parabola
Equations of Normal in Different Forms
Point of Intersection of Normals at Any Two Points on The Parabola
Relation Between \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] and \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\] if Normal at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{1}}}\mathbf{'}\] Meets the Parabola Again at \[\mathbf{'}{{\mathbf{t}}_{\mathbf{2}}}\mathbf{'}\]
Co-normal Points
Equation of The Chord of Contact of Tangents to a Parabola
Equation of the Chord of the Parabola Which is Bisected at a Given Point
Equation of the Chord Joining any Two Points on the Parabola
Diameter of a Parabola
Length of Tangent, Subtangent, Normal, Subnormal
Length of Tangent, Subtangent, Normal and Subnormal to \[{{\mathbf{y}}^{\mathbf{2}}}\mathbf{= 4ax}\,\,\mathbf{at }\,\mathbf{(a}{{\mathbf{t}}^{\mathbf{2}}}\mathbf{,2at)}\]
Pole and Polar
Definition
Standard Equation of the Ellipse
Parametric Form of The Ellipse
Special Forms of an Ellipse
Position of a Point with Respect to an Ellipse
Intersection of a Line and an Ellipse
Equations of Tangent in Different Forms
Equation of Pair of Tangents \[\mathbf{S}{{\mathbf{S}}_{\mathbf{1}}}\mathbf{=}{{\mathbf{T}}^{\mathbf{2}}}\]
Equations of Normal in Different Forms
Auxiliary Circle
Chord of Contact
Equation of Chord With Mid Point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\]
Equation of The Chord Joining Two Points on an Ellipse
Pole and Polar
Diameter of The Ellipse
Subtangent and Subnormal
Definition
Standard Equation of the Hyperbola
Conjugate Hyperbola
Special form of Hyperbola
Auxiliary Circle of Hyperbola
Parametric Equations of Hyperbola
Position of a Point With Respect to a Hyperbola
Intersection of a Line and a Hyperbola
Equations of Tangent in Different Forms
Equation of Pair of Tangents
Equations of Normal in Different Forms
Equation of Chord of Contact of Tangents Drawn from a Point to a Hyperbola
Equation of the Chord of the Hyperbola Whose mid point \[\mathbf{(}{{\mathbf{x}}_{\mathbf{1}}}\mathbf{,}\,\,{{\mathbf{y}}_{\mathbf{1}}}\mathbf{)}\] is given
Equation of The Chord Joining Two Points on The Hyperbola
Pole and Polar
Diameter of The Hyperbola
Subtangent and Subnormal of the Hyperbola
Asymptotes of a Hyperbola
Rectangular or Equilateral Hyperbola

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