Chapter 11 — Conic Sections

What are conic sections?
Conic sections (or simply conics) are the curves obtained by intersecting a plane with a double-napped right circular cone. Depending on the angle and position of the plane, the intersection is one of four types: circle, ellipse, parabola, or hyperbola.

Geometric definitions (focus–directrix properties)

• Parabola: Set of points equidistant from a fixed point (focus) and a fixed line (directrix). Eccentricity e = 1.
• Ellipse: Set of points for which the sum of distances to two fixed points (foci) is constant. 0 < e < 1.
• Hyperbola: Set of points for which the absolute difference of distances to two fixed points (foci) is constant. e > 1.
• Circle: Special case of an ellipse with equal semi-axes; eccentricity e = 0.

Analytic (equation) viewpoint
General second-degree equation: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. Classification by the discriminant Δ = B^2 - 4AC:

• Δ < 0: ellipse (if A = C and B = 0, it is a circle)
• Δ = 0: parabola
• Δ > 0: hyperbola

Standard forms (central conics aligned with axes)

• Parabola (axis along x): y^2 = 4ax. Focus (a,0), directrix x = -a, latus rectum = 4a.
• Ellipse (major axis along x): x^2/a^2 + y^2/b^2 = 1 (a > b). Foci at (±c,0) where c^2 = a^2 - b^2. Eccentricity e = c/a.
• Hyperbola (transverse axis along x): x^2/a^2 - y^2/b^2 = 1. Foci at (±c,0) where c^2 = a^2 + b^2. Eccentricity e = c/a. Asymptotes: y = ±(b/a)x.

Important geometric features: center (for ellipse/hyperbola), vertices, axis of symmetry, foci, directrices, latus rectum, eccentricity. Conics have useful reflection properties: a ray from one focus of an ellipse reflects to the other focus; rays parallel to the axis of a parabola reflect to its focus.

Why study conics?
They appear in optics, astronomy, architecture and engineering. Analytic equations let us classify and analyze shape, position, symmetry and reflective behavior, which is useful for design and physics problems in Class 11.

Definition: A circle is the locus of all points in a plane at a fixed distance (the radius) from a fixed point (the center). In conic-section language a circle is a special case of an ellipse where the two axes are equal (major axis = minor axis), so the eccentricity c/a = 0.

Geometric origin as a conic: As a conic section, a circle is obtained when a plane cuts a right circular cone in a direction perpendicular to the cone's axis (i.e., the cutting plane is parallel to the base of the cone). It is the limiting special case of an ellipse with equal semi-axes.

Key properties: All radii to points on the circle are equal. The center is equidistant from all points on the circle. A tangent at any point on a circle is perpendicular to the radius drawn to the point of contact. For a circle considered as an ellipse, the focal definition with a nonzero eccentricity does not apply (e = 0 is the limiting case).

Definition: A parabola is the locus of a point which moves such that its distance from a fixed point (the focus) is equal to its perpendicular distance from a fixed line (the directrix).

Standard forms (with parameter a > 0):

• Horizontal parabola: y² = 4ax — opens to the right. Vertex at (0,0), focus at (a,0), directrix x = -a, axis: x-axis.
• Vertical parabola: x² = 4ay — opens upward. Vertex at (0,0), focus at (0,a), directrix y = -a, axis: y-axis.

Key geometric elements: Vertex (V), Focus (F), Directrix (a line), Axis of symmetry, Latus rectum (the focal chord through the focus). For y²=4ax the latus rectum is the line segment x = a between y = -2a and y = 2a (length = 4a).

Parametric form: For y²=4ax, a general point can be written as (at², 2at). For x²=4ay, a general point is (2at, at²).

Important analytic properties:

• Distance property: For any point P(x,y) on the parabola, PF = distance to focus = distance to directrix.
• Reflection property: A ray parallel to the axis of the parabola reflects through the focus (used in reflectors and antennas).
• Conic classification: In general quadratic form Ax² + Bxy + Cy² + Dx + Ey + F = 0, the conic is a parabola if B² - 4AC = 0.

Equations of tangents and normals (for y²=4ax):

• Tangent at point (x₁,y₁) on the parabola: y y₁ = 2a(x + x₁) ("point form").
• Slope form of tangent (slope m): y = m x + a/m (m ≠ 0).
• Parametric tangent at (a t², 2 a t): t y = x + a t².
• Normal at parametric point (a t², 2 a t): t x + y = a t (t² + 2).

Why it matters (applications): Parabolas concentrate or distribute energy because of their reflection property — this is exploited in satellite dishes, headlights, parabolic microphones, and many architectural and engineering designs. Projectile motion under uniform gravity (neglecting air resistance) follows a parabolic trajectory.

Study tips / how to visualise: Start with y²=4ax with a = 1 (y²=4x). Mark the vertex (0,0), focus (1,0), directrix x = -1, and the latus rectum endpoints (1, ±2). Plot parametric points for several t values (e.g. t = -2,-1,0,1,2) and draw tangents at selected points using the parametric tangent formula to see how slopes change.

Definition (geometric): An ellipse is the set of all points in a plane for which the sum of distances to two fixed points (the foci) is constant. If the foci are F1 and F2 and P is any point on the ellipse, then PF1 + PF2 = 2a (constant).

Standard form and elements:

• Center: (0, 0) (for standard position).
• Semi-major axis = a, semi-minor axis = b, with a > b >= 0.
• If major axis is along x-axis: equation is x^2/a^2 + y^2/b^2 = 1.
• If major axis is along y-axis: equation is x^2/b^2 + y^2/a^2 = 1.
• Vertices: (±a, 0) (for horizontal major axis). Co-vertices: (0, ±b).
• Foci: (±c, 0) where c satisfies c^2 = a^2 - b^2.
• Eccentricity: e = c/a, and for an ellipse 0 < e < 1.

Important properties:

• Focal definition: PF1 + PF2 = 2a for every point P on the ellipse.
• Reflective property: A ray from one focus reflects off the ellipse and passes through the other focus (used in whispering galleries and reflecting surfaces).
• Area: A = πab.
• Length of latus rectum (through a focus, perpendicular to major axis): 2b^2/a.

General second-degree condition: A general conic Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 (with real coefficients) represents an ellipse if B^2 - 4AC < 0 and A and C have the same sign. If B != 0 the axes are rotated.

Notes on construction and concept: The ellipse can be drawn by the string-and-pins method: fix two pins at the foci, loop a string of length 2a, keep the string taut with a pencil and trace. Also the parametric form x = a cos t, y = b sin t (0 ≤ t < 2π) is useful for plotting.

Definition: A hyperbola is the set of all points in a plane such that the absolute difference of their distances from two fixed points (the foci) is a constant: |PF1 - PF2| = 2a (a > 0).

Standard forms (centered at origin):

• Transverse axis along x: x^2/a^2 - y^2/b^2 = 1 (branches open left and right).
• Transverse axis along y: y^2/a^2 - x^2/b^2 = 1 (branches open up and down).

Key elements and geometry:

• Center: (h, k). For centered form above, center is (0,0); shifted form: (x-h)^2/a^2 - (y-k)^2/b^2 = 1.
• Vertices: on the transverse axis at a units from the center: (±a, 0) for the first form.
• Foci: located at distance c from center where c^2 = a^2 + b^2. For the first form foci are (±c, 0). Eccentricity e = c/a = sqrt(1 + b^2/a^2) > 1.
• Latus rectum (focal chord through a focus): length = 2b^2/a.
• Directrices: lines x = ±a/e (for transverse axis along x). Each focus-directrix ratio yields eccentricity: distance to focus / distance to corresponding directrix = e.

Asymptotes and drawing method: For x^2/a^2 - y^2/b^2 = 1 the asymptotes are straight lines y = ±(b/a) x (shifted: (y-k) = ±(b/a)(x-h)). To sketch: draw a rectangle with corners (±a, ±b) about the center; the diagonals of this rectangle are the asymptotes. The hyperbola branches approach these diagonals.

Algebraic/analytic facts:

• Parametric forms: x = a sec t, y = b tan t (or x = a cosh u, y = b sinh u for hyperbolic functions).
• Focal parameter (semi-latus rectum) p = b^2/a; equation in focus-directrix form: distance to focus = e × distance to directrix.
• General second-degree conic Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 represents a hyperbola when B^2 - 4AC > 0.

Reflection property: A ray directed toward one focus reflects off the hyperbola and appears to come from the other focus (useful in certain optical systems).

Special case - Rectangular (equilateral) hyperbola: When a = b the asymptotes are perpendicular (slopes ±1). Example: xy = constant is an equilateral hyperbola (after rotation and scaling).

Short summary: Hyperbolas are open conics characterized by difference of distances to two foci being constant, eccentricity greater than 1, two branches with straight-line asymptotes, and important applications in physics, navigation and engineering.

Definition. For a point P on a conic, eccentricity e is the constant ratio of the distance from P to a focus (F) and the perpendicular distance from P to the corresponding directrix (l): PF = e · distance(P, l). The directrix is a fixed line associated with a focus. The value of e classifies the conic.

Classification by eccentricity.

• Ellipse: 0 ≤ e < 1. Points satisfy PF = e·(distance to directrix). When e = 0 the conic is a circle (special case).
• Parabola: e = 1. PF = distance to the directrix.
• Hyperbola: e > 1. PF = e·(distance to directrix).

Standard forms and relations (transverse axis along x-axis, center at origin):

• Ellipse: x²/a² + y²/b² = 1, with a > b > 0. Foci at (±c,0) where c² = a² − b². Eccentricity e = c/a (so 0 ≤ e < 1). Directrices: x = ± a/e.
• Parabola: y² = 4ax (vertex at origin). Focus at (a,0), directrix x = −a, eccentricity e = 1.
• Hyperbola: x²/a² − y²/b² = 1. Foci at (±c,0) where c² = a² + b². Eccentricity e = c/a (> 1). Directrices: x = ± a/e.

Alternate relations using e: For an ellipse b² = a²(1 − e²). For a hyperbola b² = a²(e² − 1).

Geometric meaning and properties. Eccentricity measures how "stretched" a conic is: smaller e (close to 0) → shape closer to circular; larger e → more open. Directrices are fixed lines such that each point keeps PF to directrix ratio equal to e. Ellipse and hyperbola have two directrices (one for each focus); parabola has one directrix (for its single focus).

How equations follow from the definition (brief). Place focus at (c,0) and directrix x = d (vertical). For a point P(x,y), PF = sqrt((x−c)² + y²) and distance to directrix = |x − d|. Using PF = e|x − d| and algebraic manipulation yields the standard second-degree equations; parameters relate so that d = a/e for centered conics.

Important notes. (1) Directrix orientation depends on axis: if transverse axis is y-axis, replace x by y in directrix equations. (2) Circle is ellipse with e = 0; the directrix moves to infinity. (3) Eccentricity is dimensionless and unique for a given conic.

Overview
A tangent to a curve at a point is the straight line that touches the curve at that point and has the same instantaneous direction (slope) as the curve there. The normal is the line perpendicular to the tangent at the same point. For an implicitly defined curve F(x,y)=0, the tangent at (x1,y1) (if the gradient is not zero) is given by the linear approximation:

Fx(x1,y1) (x - x1) + Fy(x1,y1) (y - y1) = 0,

where Fx and Fy are partial derivatives of F. The normal has slope = −1/(slope of tangent) and equation y - y1 = -1/(dy/dx|_{(x1,y1)}) (x - x1) when dy/dx is finite.

General method
1) Differentiate the conic implicitly to find dy/dx at the point.
2) Form the tangent line: y - y1 = (dy/dx)_{(x1,y1)} (x - x1).
3) Form the normal: y - y1 = -1/(dy/dx)_{(x1,y1)} (x - x1), or use geometric properties (e.g., radius ⟂ tangent for circles).

Standard results for common conics

• Circle (center (0,0), radius r): x^2 + y^2 = r^2. Tangent at (x1,y1): x x1 + y y1 = r^2. The normal is the radius through (x1,y1): y - y1 = (y1/x1)(x - x1) (unless x1=0 vertical).
• Parabola y^2 = 4ax: point (x1,y1) satisfies y1^2 = 4ax1. Tangent: y y1 = 2a (x + x1). Parametric form: for t, point (a t^2, 2 a t) and tangent: t y = x + a t^2. Normal (parametric): y - 2 a t = -t (x - a t^2), or y = -t x + a t^3 + 2 a t.
• Parabola x^2 = 4ay: tangent at (x1,y1): x x1 = 2a (y + y1).
• Ellipse x^2/a^2 + y^2/b^2 = 1: tangent at (x1,y1): (x x1)/a^2 + (y y1)/b^2 = 1. Parametric point (a cos θ, b sin θ) gives tangent: (x cos θ)/a + (y sin θ)/b = 1. Slope form: y = m x ± sqrt(a^2 m^2 + b^2).
• Hyperbola x^2/a^2 - y^2/b^2 = 1: tangent at (x1,y1): (x x1)/a^2 - (y y1)/b^2 = 1. Slope form: y = m x ± sqrt(a^2 m^2 - b^2) (real tangents only if a^2 m^2 ≥ b^2).

Nature and applications
Tangent and normal lines are fundamental in geometry and calculus and have many real-life applications: optics (reflection laws use the normal), engineering (normal force directions, contact forces), road/rail design (tangents to curves for smooth transitions), computer graphics (shading and reflection rely on normals), and reflector design (parabolic dishes use tangent/normal for directing waves to the focus).

Definition — Chord: A chord of a conic is a straight line segment joining two points on the conic. If the points are P(x1,y1) and Q(x2,y2), the equation of the chord (the line through P and Q) can be written in point-slope or two-point form.

Focal Chord: A focal chord is a chord that passes through a focus of the conic. A special focal chord perpendicular to the principal axis is called the latus rectum.

Parabola (detailed)
Take the standard parabola y^2 = 4ax with parametric point for parameter t: (at^2, 2at).

Chord joining two parametric points t1 and t2 has equation (derived by eliminating parameters):

y(t1 + t2) = 2x + 2a t1 t2.

If this chord passes through the focus (a, 0) substitute x = a, y = 0 to get

0 = 2a + 2a t1 t2 ⇒ t1 t2 = −1.

So for a parabola, the parameters of the endpoints of any focal chord are reciprocals with opposite sign: if one end is t, the other is −1/t.

Latus rectum of the parabola: the focal chord perpendicular to axis; for y^2 = 4ax its endpoints are (a, ±2a) and its length = 4a.

Ellipse and Hyperbola (brief)
Standard ellipse: x^2/a^2 + y^2/b^2 = 1, with foci at (±c,0), c^2 = a^2 − b^2. A focal chord is any chord through (c,0) (or (−c,0)). The latus rectum through focus (c,0) is the chord whose x-coordinate is c: substitute x = c in ellipse to get y = ±(b^2/a). Thus endpoints of the latus rectum: (c, ±b^2/a) and its length = 2 b^2 / a.

The same latus-rectum length formula (2 b^2 / a) holds for the standard hyperbola x^2/a^2 − y^2/b^2 = 1 with focus at (±c,0) where c^2 = a^2 + b^2; the latus-rectum endpoints are (c, ± b^2 / a).

How to find a focal chord in general: Write the equation of a line through a focus (e.g. y = m(x − c) for focus (c,0)), substitute into the conic equation to obtain a quadratic in x (or parameter). The roots correspond to the x-coordinates (or parameters) of the intersection points; impose the condition that the line passes through the focus (already used) and solve for the intersection points. For parabola, this leads to the simple param condition t1 t2 = −1.

Key geometric facts to remember:

• Chord = straight line joining two points on conic.
• Focal chord = chord passing through a focus.
• Parabola: focal-chord parameter condition t1 t2 = −1.
• Parabola latus rectum length = 4a (for y^2 = 4ax), endpoints (a, ±2a).
• Ellipse/hyperbola latus rectum length = 2 b^2 / a, endpoints at (±c, ± b^2 / a) depending on focus chosen.

These properties are used frequently in problems involving reflection, focal properties, and chord geometry (midpoints, intercepts, etc.).

General second-degree equation (conic):
The most general equation of a conic in two variables x and y is
Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0, where A, B, C, D, E, F are real constants and not all A, B, C are zero.

Classification (using the quadratic part):
Define the discriminant for classification as Δ1 = B² − AC.

• If Δ1 < 0: the conic is an ellipse (special case A = C, B = 0 gives a circle).
• If Δ1 = 0: the conic is a parabola.
• If Δ1 > 0: the conic is a hyperbola.

Center and translation:
If the conic has a center (non-parabolic cases), the center (x₀, y₀) is found by solving the linear system obtained from first derivatives:

Ax₀ + By₀ + D = 0
Bx₀ + Cy₀ + E = 0

Equivalently, in matrix form: [ [A, B], [B, C] ] [x₀, y₀ ]^T = -[D, E]^T.

Rotation to remove the xy-term:
If B ≠ 0, the conic is generally rotated. Use a rotation of axes by θ where

tan 2θ = 2B / (A − C).

Choosing θ that satisfies this eliminates the xy-term; then translate to the center (if any) to get the canonical form.

Degenerate cases:
The conic may reduce to a pair of lines, a single line, a point, or be empty. A useful determinant test for degeneracy is the 3×3 determinant

Δ = det [[A, B, D], [B, C, E], [D, E, F]]. If Δ = 0 the conic is degenerate (possibly pair of lines or a point); if Δ ≠ 0 it is non-degenerate.

Canonical forms after suitable rotation and translation:

• Ellipse: (x')²/a² + (y')²/b² = 1
• Circle: (x')² + (y')² = r² (special ellipse)
• Parabola: (y')² = 4a x' or (x')² = 4a y'
• Hyperbola: (x')²/a² − (y')²/b² = 1 (or swapped signs)

Procedure to classify any given second-degree equation:

1. Write in standard form Ax² + 2Bxy + Cy² + 2Dx + 2Ey + F = 0.
2. Compute Δ1 = B² − AC to identify ellipse/parabola/hyperbola.
3. If needed, find center by solving Ax + By + D = 0, Bx + Cy + E = 0 and translate coordinates.
4. If B ≠ 0, compute θ from tan 2θ = 2B/(A − C) to rotate axes and remove the xy-term, then reduce to canonical form.
5. Check the 3×3 determinant Δ to detect degenerate cases.

Overview: The general second degree (conic) equation is

Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F = 0

Methods of transformation (translation and rotation of axes) are used to simplify this equation to a standard form (circle, ellipse, parabola, hyperbola) so geometric properties (centre, axes, foci, directrices) can be read off easily.

1. Translation (shift of origin)
Purpose: remove linear terms (2Dx + 2Ey) by shifting the origin to the conic's center (if it exists).
If (h,k) is the new origin, set x = X + h, y = Y + k and substitute. For a conic with a center, (h,k) solves the linear system

Ah + Bk + D = 0
Bh + Ck + E = 0

or by Cramer's rule

h = (B E - C D)/(A C - B^2),  k = (B D - A E)/(A C - B^2)

where AC - B^2 ≠ 0. After translation, complete the squares in X,Y to reach a centred standard form.

2. Rotation of axes
Purpose: remove the cross-term 2Bxy when B ≠ 0 (tilted conics). Rotate axes by angle θ using the relations

x = X cosθ - Y sinθ
y = X sinθ + Y cosθ

Choose θ so that the coefficient of XY in the new equation is zero. The required angle satisfies

tan(2θ) = 2B / (A - C)

After rotation the quadratic part becomes diagonal (no XY term): A'X^2 + C'Y^2 + ... = 0. A' and C' are the eigenvalues of the symmetric matrix [[A, B],[B, C]].

3. Combined procedure to obtain standard form

1. If B ≠ 0, rotate axes to eliminate the XY term (use tan2θ = 2B/(A−C)).
2. Translate to the centre (if conic is non-parabolic) to remove linear terms; solve for (h,k) as above.
3. Complete the square to obtain one of the standard forms:
   • Ellipse: (X^2/a^2) + (Y^2/b^2) = 1
   • Parabola: Y^2 = 4aX (or X^2 = 4aY)
   • Hyperbola: (X^2/a^2) - (Y^2/b^2) = 1

4. Classification using the discriminant
For the general form Ax^2 + 2Bxy + Cy^2 + 2Dx + 2Ey + F = 0, use the invariant

Δ = B^2 - A C

to classify the conic (assuming non-degenerate):

• Δ < 0: ellipse (circle if A = C and B = 0)
• Δ = 0: parabola
• Δ > 0: hyperbola

5. Matrix viewpoint (compact)
Write the quadratic form as [x y]Q[x;y] + 2[p]^T[x;y] + F = 0, where Q = [[A,B],[B,C]] and p = [D,E]. Diagonalizing Q by an orthogonal matrix (rotation) removes cross-terms; translation eliminates the linear term. Eigenvalues of Q give principal-axis coefficients.

Remarks and tips:

• Parabolas have no centre (AC - B^2 = 0), so translation to a centre is not possible; instead rotate first (if needed) and then complete the square for the linear direction.
• Always try to remove the XY-term first (by rotation) unless B = 0. After diagonalization, translation is straightforward.
• Use completing-the-square carefully after coordinate substitutions to identify a, b, c, eccentricity e, asymptotes, foci.

In Class 11 Conic Sections, the chapter 'Applications and Problems' focuses on using the geometric and analytic properties of parabola, ellipse and hyperbola to model, solve and interpret real-world and exam-style problems. Typical tasks include: identifying the conic from a locus or condition; deriving equations from focus-directrix or distance conditions; finding tangents, normals, chords, foci and vertices; using parametric forms for computations; and applying reflective and distance properties to practical designs.

• Approach to problems: (1) Identify which conic fits the condition (use eccentricity e or distance relations). (2) Choose a convenient coordinate system (translate/rotate if necessary). (3) Use standard equation, focus-directrix property or parametric form for algebraic work. (4) Use tangency/condition formulas for lines and chords. (5) Interpret results in context (distances, reflection, geometry).
• Common problem types: find equation from focus & directrix; find conic from foci and major/minor lengths; locate tangent or normal at a point; length of latus rectum and focal parameter; intersection with lines and circles; reflective property problems (optics, antennas); construction/design tasks (paths, orbits).
• Physical/Engineering use: Parabolas for antennas, headlights and bridges; ellipses for planetary orbits and whispering galleries; hyperbolas for navigation systems and asymptotic approximations.

Mastering these problems requires fluency with standard forms, definitions (focus, directrix, eccentricity), parametric coordinates, and algebraic manipulations (squaring distance relations, completing the square, using symmetry). Visualizing graphs (foci, directrices, axes, asymptotes and reflective rays) is crucial to understanding and checking solutions.