Circles and Conic Sections (Parabola, Ellipse, Hyperbola) are crucial topics in Coordinate Geometry for the NDA 2 2025 Mathematics paper. These concepts require a strong understanding of their standard equations, properties, and various forms. Mastering them is essential for solving problems related to geometry, loci, and curves.
1. Circles:
- Standard Equation of a Circle:
- Center at origin (0,0) and radius r: x2+y2=r2.
- Center at (h,k) and radius r: (x−h)2+(y−k)2=r2.
- General Equation of a Circle: x2+y2+2gx+2fy+c=0.
- Center is (−g,−f) and radius r=g2+f2−c
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- Center is (−g,−f) and radius r=g2+f2−c
- Conditions for a Circle:
- Coefficient of x2 must be equal to the coefficient of y2 (and non-zero).
- No xy term.
- Equation of Tangent and Normal:
- At a given point on the circle.
- With a given slope.
- Chord of Contact: Equation of the chord joining the points of contact of two tangents from an external point.
2. Conic Sections:
Conic sections are curves formed by the intersection of a plane with a double-napped cone.
- Parabola: A set of all points that are equidistant from a fixed point (focus) and a fixed line (directrix).
- Standard Forms: y2=4ax, y2=−4ax, x2=4ay, x2=−4ay.
- Key Properties: Vertex, Focus, Directrix, Latus Rectum, Axis.
- Ellipse: A set of all points for which the sum of the distances from two fixed points (foci) is constant.
- Standard Forms: a2x2+b2y2=1 (major axis along x-axis), b2x2+a2y2=1 (major axis along y-axis).
- Key Properties: Foci, Vertices, Co-vertices, Major and Minor axes, Eccentricity (e=c/a, where c2=a2−b2), Latus Rectum. Note 0<e<1.
- Hyperbola: A set of all points for which the absolute difference of the distances from two fixed points (foci) is constant.
- Standard Forms: a2x2−b2y2=1 (transverse axis along x-axis), a2y2−b2x2=1 (transverse axis along y-axis).
- Key Properties: Foci, Vertices, Transverse and Conjugate axes, Eccentricity (e=c/a, where c2=a2+b2), Latus Rectum, Asymptotes. Note e>1.
- Degenerate Conics: Pair of lines, point.
Preparation Strategy:
- Memorize Standard Equations: Crucial for quickly identifying the type of conic and its parameters.
- Understand Properties: Know the definitions and significance of focus, directrix, eccentricity, axes, latus rectum for each conic.
- Practice Drawing/Visualizing: A rough sketch helps in understanding the geometry of the problem.
- Solve Variety of Problems: Practice problems involving finding equations, finding tangent/normal, properties, and applications.
- Focus on Key Differences: Be clear about the distinct properties and equations of each conic.
Mastering Circles and Conic Sections is essential for securing marks in the Coordinate Geometry section of the NDA 2 2025 Mathematics paper.
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