Conic Sections revision notes

A cone is generated by rotating a line $m$ (generator) around a fixed vertical line $l$ (axis) intersecting at a fixed point $V$ (vertex) at a constant angle $\alpha$. This generates a double-napped right circular hollow cone. The intersection of a plane with this cone yields various conic sections.

Let $\beta$ be the angle made by the intersecting plane with the vertical axis of the cone.

Standard Conic Sections (Plane cuts a nappe, not the vertex)

• Circle: When $\beta = 90^\circ$. The section is perfectly circular.
• Ellipse: When $\alpha < \beta < 90^\circ$. The plane cuts entirely across one nappe.
• Parabola: When $\beta = \alpha$. The plane is parallel to a generator and cuts one nappe.
• Hyperbola: When $0 \le \beta < \alpha$. The plane cuts through both nappes, yielding two disjoint curves.

Degenerated Conic Sections (Plane cuts at the vertex $V$)

• Point: When $\alpha < \beta \le 90^\circ$. A degenerated circle/ellipse.
• Straight Line: When $\beta = \alpha$. The plane contains a generator. A degenerated parabola.
• Pair of Intersecting Straight Lines: When $0 \le \beta < \alpha$. A degenerated hyperbola.

A circle is the set of all points in a plane equidistant (radius, $r$) from a fixed point (centre).

• Standard Equation (Centre at origin): $x^2 + y^2 = r^2$.
• Equation with Centre $(h, k)$: $(x - h)^2 + (y - k)^2 = r^2$.

• Diametric Form (JEE Advanced Topic): If $(x_1, y_1)$ and $(x_2, y_2)$ are endpoints of a diameter, the equation is $(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$.
• Position of a Point: For a point $P(x_1, y_1)$ and circle $S \equiv x^2 + y^2 - r^2 = 0$, $P$ lies outside, on, or inside the circle if $S_1 = x_1^2 + y_1^2 - r^2$ is $>0$, $=0$, or $<0$ respectively.

A parabola is the locus of a point in a plane equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.

• Axis: The line through the focus perpendicular to the directrix. The parabola is symmetric about its axis.
• Vertex: The point of intersection of the parabola with its axis.
• Latus Rectum: A line segment perpendicular to the axis, passing through the focus, with endpoints on the parabola. For $y^2 = 4ax$, Length $= 4a$.

Standard Orientations

1. Rightward Opening ($y^2 = 4ax$, $a > 0$): Focus $(a, 0)$, Directrix $x = -a$, Axis $y = 0$, Vertex $(0,0)$. Latus rectum ends: $(a, 2a)$ and $(a, -2a)$.
2. Leftward Opening ($y^2 = -4ax$, $a > 0$): Focus $(-a, 0)$, Directrix $x = a$, Axis $y = 0$.
3. Upward Opening ($x^2 = 4ay$, $a > 0$): Focus $(0, a)$, Directrix $y = -a$, Axis $x = 0$.
4. Downward Opening ($x^2 = -4ay$, $a > 0$): Focus $(0, -a)$, Directrix $y = a$, Axis $x = 0$.

• Parametric Form (JEE Advanced Topic): For $y^2 = 4ax$, any point on the parabola is $(at^2, 2at)$. JEE Tip Always use parametric coordinates for locus problems involving chords or tangents to minimize variables.
• Focal Chord Property (JEE Advanced Topic): If the endpoints of a focal chord have parameters $t_1$ and $t_2$, then $t_1 t_2 = -1$.

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (foci) is a constant ($2a$). This constant is always greater than the distance between the foci ($2c$).

• Centre: Midpoint of the line segment joining the foci.
• Major Axis: Line segment through the foci. Length $= 2a$. Endpoints are vertices.
• Minor Axis: Line segment through the centre, perpendicular to the major axis. Length $= 2b$.
• Relation between $a, b, c$: $c^2 = a^2 - b^2$, meaning $c = \sqrt{a^2 - b^2}$.
• Eccentricity ($e$): Ratio of the distance from the centre to a focus ($c$) to the distance from the centre to a vertex ($a$). $e = c/a$. Thus, $c = ae$. For an ellipse, $e < 1$.
• Latus Rectum: Length $= 2b^2/a$. Endpoints for horizontal ellipse are $(\pm ae, \pm b^2/a)$.

Standard Orientations

1. Horizontal Ellipse ($\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b$): Major axis along x-axis. Foci $(\pm c, 0) = (\pm ae, 0)$. Vertices $(\pm a, 0)$. Domain: $-a \le x \le a$. Range: $-b \le y \le b$.
2. Vertical Ellipse ($\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$, where $a > b$): Major axis along y-axis. Foci $(0, \pm c) = (0, \pm ae)$. Vertices $(0, \pm a)$. JEE Tip Do not blindly memorize "a is under x". $a$ is always the semi-major axis. Check which denominator is larger!

• Parametric Form (JEE Advanced Topic): Any point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ is $(a \cos \theta, b \sin \theta)$, where $\theta$ is the eccentric angle.

A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points (foci) is a constant ($2a$).

• Transverse Axis: The line segment through the foci. Length $= 2a$. Intersects the hyperbola at vertices.
• Conjugate Axis: The line segment through the centre, perpendicular to the transverse axis. Length $= 2b$.
• Relation between $a, b, c$: $b^2 = c^2 - a^2$, meaning $c^2 = a^2 + b^2$.
• Eccentricity ($e$): $e = c/a$. Since $c \ge a$, $e > 1$. Foci are at distance $ae$ from the centre.
• Latus Rectum: Length $= 2b^2/a$.

Standard Orientations

1. Horizontal Hyperbola ($\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$): Transverse axis along x-axis. Foci $(\pm c, 0) = (\pm ae, 0)$. Vertices $(\pm a, 0)$. Domain: $x \le -a$ or $x \ge a$.
2. Vertical Hyperbola ($\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$): Transverse axis along y-axis. Foci $(0, \pm c) = (0, \pm ae)$. Vertices $(0, \pm a)$. JEE Tip The positive term dictates the transverse axis, not the larger denominator!

• Equilateral Hyperbola: A hyperbola where $a = b$. Equation: $x^2 - y^2 = a^2$.
• Rectangular Hyperbola (JEE Advanced Topic): The standard equilateral hyperbola rotated by $45^\circ$, giving $xy = c^2$. Parametric form: $(ct, c/t)$.

General Equation of Conics (JEE Advanced): $S \equiv ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$

• $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$.
• If $\Delta = 0$: Pair of straight lines.

Tangent Equations (JEE Advanced):

• Parabola ($y^2=4ax$): $y = mx + a/m$
• Ellipse: $y = mx \pm \sqrt{a^2m^2 + b^2}$
• Hyperbola: $y = mx \pm \sqrt{a^2m^2 - b^2}$

• The fundamental relation $PF = e \cdot PM$ (distance to focus = $e \times$ distance to directrix) is strictly valid for all conics.
• Standard equations of conics ($\frac{x^2}{a^2} \pm \frac{y^2}{b^2} = 1$ or $y^2 = 4ax$) only apply when the axes of symmetry perfectly align with the coordinate axes ($x$-axis and $y$-axis) and the center/vertex is at the origin $(0,0)$. If the axes are shifted or rotated, the equations become much more complex involving $(x-h), (y-k)$, and $xy$ terms.

• Traps regarding Ellipse major/minor axis: Students often assume $a$ is always the $x$-denominator and $b$ is the $y$-denominator. CORRECT UNDERSTANDING: In NCERT standard notation, $a$ is ALWAYS the semi-major axis (the larger value). So if the $y$-denominator is larger, the equation is effectively $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ and the major axis lies along the $y$-axis.
• Traps regarding Hyperbola transverse axis: Students often think the larger denominator dictates the transverse axis (like the ellipse). CORRECT UNDERSTANDING: For a hyperbola, the positive term determines the transverse axis. E.g., in $\frac{y^2}{25} - \frac{x^2}{100} = 1$, the transverse axis is along the $y$-axis, even though 100 > 25.

1. Parametric Locus Problems: Using $(at^2, 2at)$ or $(a\cos\theta, b\sin\theta)$ to find the locus of midpoints of chords or intersection of tangents.
2. Common Tangents: Finding the equation of a line tangent to both a circle and a parabola, or an ellipse and a hyperbola.
3. Focal Chord Properties: Specifically using $t_1 t_2 = -1$ for parabola focal chords, and proving the semi-latus rectum is the harmonic mean of the segments of a focal chord.
4. Director Circles: The locus of intersection of perpendicular tangents. For $\frac{x^2}{a^2} \pm \frac{y^2}{b^2} = 1$, it's $x^2 + y^2 = a^2 \pm b^2$.

Derivation of Standard Parabola ($y^2 = 4ax$)

1. Let focus $F(a, 0)$ and directrix line $l \equiv x + a = 0$.
2. Let $P(x,y)$ be any point on the parabola. Draw $PB \perp l$. The coordinates of $B$ are $(-a, y)$.
3. By definition, distance to focus = distance to directrix: $PF = PB$.
4. Using distance formula: $\sqrt{(x-a)^2 + (y-0)^2} = \sqrt{(x-(-a))^2 + (y-y)^2}$.
5. Squaring both sides: $(x-a)^2 + y^2 = (x+a)^2$.
6. Expanding: $x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2 \Rightarrow y^2 = 4ax$.

Practical Application Problem (Parabolic Mirror / Beam Deflection)

Setup: If a physical structure (mirror, bridge cable, rod) forms a conic section, set the vertex at the origin $(0,0)$ to simplify equations.

Application Problem (Sliding Rod forming an Ellipse)

• Scenario: A rod $AB$ of length 15cm rests between axes (A on x-axis, B on y-axis). Point $P(x,y)$ is 6cm from A. Locus of P?
• Solution: Let angle with x-axis be $\theta$. $AP=6$, $PB=9$. $x = 9 \cos\theta$, $y = 6 \sin\theta$. Using $\sin^2\theta + \cos^2\theta = 1$, we get $\frac{x^2}{81} + \frac{y^2}{36} = 1$. The locus is an ellipse.