Analytical Geometry Questions for NDA

Straight Lines — Core Formulas and Conditions

A straight line is the simplest curve in coordinate geometry. NDA regularly asks about slope, angle between lines, distance formulas, and position of a point relative to a line. The key skill is picking the right form of the line equation based on what information is given.

• Slope m = tan(theta) where theta is angle with positive x-axis
• Two lines are parallel if m1 = m2, perpendicular if m1 x m2 = -1
• Distance from point (x1, y1) to line ax + by + c = 0 is |ax1 + by1 + c| / sqrt(a^2 + b^2)
• Angle bisector of two lines: use the distance-from-line formula set equal for both lines
• Foot of perpendicular and reflection of a point are frequent NDA question types
• Three lines are concurrent if the determinant of their coefficients is zero

Circle — Standard Forms and Key Results

Circle questions in NDA focus on finding the equation, checking if a line is tangent, length of tangent from an external point, and common chords. The general equation approach is used when center and radius are not directly obvious.

• Standard form: center (0,0), radius r => x^2 + y^2 = r^2
• General form: x^2 + y^2 + 2gx + 2fy + c = 0, center = (-g, -f), radius = sqrt(g^2 + f^2 - c)
• Condition for tangency: distance from center to line = radius
• Length of tangent from external point P(x1,y1) = sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c)
• Chord of contact from external point (x1,y1): T = 0 means xx1 + yy1 = r^2
• Two circles intersect, touch externally, touch internally based on relation between d and r1, r2

Parabola — Standard Form and Properties

NDA consistently asks 2-3 questions on parabola covering focus, directrix, tangent equations, and parametric form. Always identify which standard form applies before solving. The most tested is y^2 = 4ax.

• Standard parabola: y^2 = 4ax, focus = (a,0), directrix: x = -a, vertex = (0,0)
• Axis is along x-axis for y^2=4ax, along y-axis for x^2=4ay
• Parametric point on y^2=4ax is (at^2, 2at)
• Tangent at point (at^2, 2at): ty = x + at^2
• Normal at parametric point t: y = -tx + 2at + at^3
• Focal chord: if one end has parameter t, other end has parameter -1/t

Ellipse and Hyperbola — Key Results Only

NDA tests basic properties of ellipse and hyperbola — mostly identifying center, foci, eccentricity, and directrix. Deep derivations are not expected. Focus on remembering the comparison between the two: ellipse has e less than 1, hyperbola has e greater than 1.

• Ellipse: x^2/a^2 + y^2/b^2 = 1 where a > b, foci at (+-ae, 0), b^2 = a^2(1-e^2)
• Hyperbola: x^2/a^2 - y^2/b^2 = 1, foci at (+-ae, 0), b^2 = a^2(e^2-1)
• For ellipse: sum of focal distances = 2a (constant), that is why it is an ellipse
• Asymptotes of hyperbola: y = +-(b/a)x — these pass through center
• Rectangular hyperbola: xy = c^2, parametric point (ct, c/t)
• Eccentricity: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1

• Confusing the general circle form: students write center as (g,f) instead of (-g,-f). Remember the negative sign always.
• For parabola y^2=4ax, students forget that the focus is (a,0) not (4a,0). Extract a first, then find focus.
• Mixing up perpendicular condition: m1*m2 = -1 only when neither line is vertical. If one line is vertical (undefined slope) the other must be horizontal.
• In ellipse, confusing which is a and which is b. a is always the larger denominator. b^2 = a^2(1-e^2) not the other way.
• Forgetting to check if a point is inside or outside the circle before applying tangent formula. Tangent from internal point is not real.

• SPFDN for Parabola: Standard form, Parametric point, Focus, Directrix, Normal — recall these 5 in order for any parabola question.
• Eccentricity ladder: 0 (circle) → less than 1 (ellipse) → 1 (parabola) → more than 1 (hyperbola). Think of it as a door opening wider and wider.
• For circle general form, think of g and f as enemies of the center — they flip sign. Center = (-g, -f).
• Focal chord of parabola mnemonic: t and -1/t are always together like a seesaw — if one end is t, the other is the negative reciprocal.

• NDA papers typically have 3-4 questions on straight lines, 2-3 on circles, 2 on parabola, and 1 each on ellipse and hyperbola. Straight lines and circles together account for over half the section.
• Most questions are one-step formula substitution. If a question is taking more than 2 minutes, re-read it — you are likely missing a shortcut or using the wrong form.
• Locus problems appear in almost every NDA paper. The trick is to assume the moving point as (h,k), apply the given condition, then replace h with x and k with y.
• For tangent problems, NDA often gives a line and asks if it is tangent to a curve — just apply the condition (discriminant = 0 or distance = radius) rather than solving the full system.
• Parametric form of parabola (at^2, 2at) saves time in chord and normal questions. Practice substituting parametric coordinates directly.