Coordinate GeometryHigh Yield (2-3 Qs in JEE)

Parabola & Conics Formulas

Standard parabola y² = 4ax, parametric form, tangent, normal in slope form, focal chords, and reflection property.

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Concept Diagram

Standard Parabola & Conic Geometry

Standard parabola y² = 4ax with focus S(a,0), directrix x = -a, vertex (0,0), and latus rectum length 4a.

y^2 = 4ax \implies \text{Focus: } (a, 0), \quad \text{Directrix: } x = -a, \quad \text{Latus Rectum: } 4a

Diagram Reference:Standard Parabola Parameters

Standard horizontal parabola opened to the right with vertex at origin.

Examiner Tip: Parametric coordinates are (at², 2at) with parameter t.

y = mx + \frac{a}{m} \quad (m \ne 0)

Equation of tangent to y² = 4ax with slope m.

Examiner Tip: Point of contact is (a/m², 2a/m).

y = mx - 2am - am^3

Equation of normal with slope m to y² = 4ax.

Examiner Tip: Three normals can be drawn from a point (h, k) to a parabola. Sum of slopes m₁ + m₂ + m₃ = 0.

t_1 t_2 = -1, \quad \text{Length of Focal Chord} = a\left(t + \frac{1}{t}\right)^2 = 4a\csc^2\theta

Parametric relation connecting endpoints of a chord passing through the focus (a, 0).

Examiner Tip: Minimum length of any focal chord is the latus rectum (4a) at θ = 90°.

ty = x + at^2

Parametric tangent to parabola at point P(at², 2at).

Examiner Tip: Intersection of tangents at t₁ and t₂ is (a t₁ t₂, a(t₁ + t₂)).

y + tx = 2at + at^3, \quad t_2 = -t_1 - \frac{2}{t_1}

Parametric normal at (at₁², 2at₁) intersects the parabola again at parameter t₂.

Examiner Tip: Intersection of normals at t₁ and t₂ is (2a + a(t₁² + t₂² + t₁t₂), -at₁t₂(t₁ + t₂)).

m_1 + m_2 + m_3 = 0, \quad y_1 + y_2 + y_3 = 0, \quad x_1 + x_2 + x_3 = 2(h - 2a)

Properties of the 3 points whose normals concur at (h, k).

Examiner Tip: Centroid of the triangle formed by 3 co-normal points has y-coordinate 0 (always lies on the axis of parabola).

x = -a \quad (\text{The Directrix})

Locus of point of intersection of two mutually perpendicular tangents to parabola y² = 4ax.

Examiner Tip: Tangents drawn from any point on the directrix always meet at right angles (90°).

Frequently tested misconceptions in Parabola & Conics