NDA Analytical Geometry: Lines, Conics, Planes and Spheres

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Rectangular Cartesian coordinates; distance formula; equations of a line in various forms; angle between lines; distance of a point from a line; circle in standard and general form; standard parabola, ellipse and hyperbola; eccentricity and axes. In 3D: points, distance, direction cosines / ratios, line and plane equations, angles, sphere.

Rectangular Cartesian coordinates; distance formula; equations of a line in various forms; angle between lines; distance of a point from a line; circle in standard and general form; standard parabola, ellipse and hyperbola; eccentricity and axes. In 3D: points, distance, direction cosines / ratios, line and plane equations, angles, sphere.

Algebra, quadratic manipulation, basic trigonometry and coordinate plotting.

• Slope and intercept
• Conic section and eccentricity
• Direction ratios and direction cosines
• Plane normal

• 2D distance: sqrt[(x2-x1)^2+(y2-y1)^2]
• Slope through two points: (y2-y1)/(x2-x1), x2!=x1
• Point-slope line: y-y1=m(x-x1)
• Point to Ax+By+C=0 distance: |Ax0+By0+C|/sqrt(A^2+B^2)
• Circle: (x-h)^2+(y-k)^2=r^2
• Parabola: y^2=4ax or x^2=4ay in standard orientation
• Ellipse: x^2/a^2+y^2/b^2=1, a>b>0; e=sqrt(1-b^2/a^2)
• Hyperbola: x^2/a^2-y^2/b^2=1; e=sqrt(1+b^2/a^2)
• 3D distance: sqrt[(x2-x1)^2+(y2-y1)^2+(z2-z1)^2]
• Sphere: (x-a)^2+(y-b)^2+(z-c)^2=r^2

• Slope formula excludes vertical lines; use x=constant instead.
• Distance-to-line formula assumes standard line coefficients not all zero.
• Conic formulas depend on orientation and positive semi-axis parameters.
• Direction cosines l,m,n satisfy l^2+m^2+n^2=1.
• Angle formulas must use correct line / plane direction data.

1. Line through (2,3) with slope 4: y-3=4(x-2).
2. Circle x^2+y^2-6x+4y-12=0 becomes (x-3)^2+(y+2)^2=25 by completing squares.
3. Distance between (1,2,3) and (4,6,3) is sqrt(9+16)=5.

• Treating a vertical line as having finite slope.
• Forgetting absolute value in point-to-line distance.
• Mixing ellipse and hyperbola eccentricity relations.
• Completing the square incorrectly in a circle.
• Using direction ratios as direction cosines without normalisation when the formula requires cosines.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

Sketch the geometry when possible. Convert equations to a recognisable standard form before extracting centre, radius, axes or eccentricity. In 3D, separate point, line and plane objects carefully. Choose vector methods only if they reduce algebra, not by habit.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.