Circle, Parabola, Ellipse and Hyperbola

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your ECAT exam.

These four curves are the conic sections — formed when a plane cuts a double cone. The type of curve depends on the angle between the plane and the cone’s axis.

The Circle: $$(x - h)^2 + (y - k)^2 = r^2$$ Centre $(h, k)$, radius $r$.

General form: $x^2 + y^2 + 2gx + 2fy + c = 0$ Centre: $(-g, -f)$, Radius: $\sqrt{g^2 + f^2 - c}$ (real if $g^2 + f^2 > c$).

The Parabola: The locus of points equidistant from a fixed point (focus) and a fixed line (directrix).

Standard forms (vertex at origin):

• $y^2 = 4ax$ (opens right), focus $(a, 0)$, directrix $x = -a$
• $x^2 = 4ay$ (opens up), focus $(0, a)$, directrix $y = -a$
• $y^2 = -4ax$ (opens left), $x^2 = -4ay$ (opens down)

The Ellipse: The locus of points where the sum of distances to two fixed points (foci) is constant.

Standard form: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (major axis horizontal if $a > b$)

• Foci: $(\pm c, 0)$ where $c^2 = a^2 - b^2$
• Eccentricity: $e = c/a$ ($0 < e < 1$ for ellipse)
• Latus rectum: $2b^2/a$

The Hyperbola: The locus of points where the absolute difference of distances to two fixed points (foci) is constant.

Standard form: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

• Foci: $(\pm c, 0)$ where $c^2 = a^2 + b^2$
• Eccentricity: $e = c/a$ ($e > 1$ for hyperbola)
• Asymptotes: $y = \pm \frac{b}{a}x$

⚡ ECAT exam tips:

• For circle: centre $(-g, -f)$ — not $(g, f)$
• Parabola $y^2 = 4ax$: focal length = $a$; focal parameter (latus rectum) = $4a$
• Ellipse: sum of distances to foci = $2a$ (constant)
• Hyperbola: absolute difference of distances to foci = $2a$ (constant)
• $e < 1$: ellipse; $e = 1$: parabola; $e > 1$: hyperbola

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🟡 Standard — Regular Study (2d–2mo)

For ECAT students who want genuine understanding of conic sections.

Parametric Forms:

Curve	Parametric equations
Circle $x^2 + y^2 = a^2$	$x = a\cos\theta$, $y = a\sin\theta$
Parabola $y^2 = 4ax$	$x = at^2$, $y = 2at$
Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	$x = a\cos\theta$, $y = b\sin\theta$
Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	$x = a\sec\theta$, $y = b\tan\theta$

Tangents — Standard Results:

Curve	Point	Tangent equation
Circle $x^2+y^2=a^2$	$(x_1,y_1)$	$xx_1 + yy_1 = a^2$
Parabola $y^2=4ax$	$(at^2, 2at)$	$ty = x + at^2$
Ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	$(x_1,y_1)$	$\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$
Hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$(x_1,y_1)$	$\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$

Condition for Tangency:

For $y = mx + c$ to be tangent to:

• Circle $x^2 + y^2 = a^2$: $c^2 = a^2(1 + m^2)$
• Parabola $y^2 = 4ax$: $c = a/m$
• Ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $c^2 = a^2m^2 + b^2$
• Hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $c^2 = a^2m^2 - b^2$

Normals:

For parabola $y^2 = 4ax$ at point $(at^2, 2at)$: Normal: $y = -tx + 2at + at^3$

⚡ Common student mistakes:

1. Forgetting that $c^2$ must be positive in the tangency condition — a real tangent exists only when the right-hand side is positive
2. Confusing the parametric form of ellipse ($x=a\cos\theta$, $y=b\sin\theta$) with circle ($x=a\cos\theta$, $y=a\sin\theta$)
3. Getting the asymptote of hyperbola wrong: $y = \pm \frac{b}{a}x$ — check the sign carefully

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for ECAT mastery of all conic sections.

Directrix Property and Eccentricity:

Every conic section can be defined by the ratio of distance from focus to distance from directrix:

• If $e = 1$: parabola
• If $e < 1$: ellipse
• If $e > 1$: hyperbola

For the parabola $y^2 = 4ax$: focus is $(a, 0)$, directrix is $x = -a$. For any point $(x,y)$ on the parabola: $\sqrt{(x-a)^2 + y^2} = x + a$ (distance to focus = distance to directrix). Squaring: $x^2 - 2ax + a^2 + y^2 = x^2 + 2ax + a^2$. So $y^2 = 4ax$. ✓

Chords and Latus Rectum:

The latus rectum is the chord through the focus perpendicular to the axis. For $y^2 = 4ax$, the latus rectum has endpoints $(a, 2a)$ and $(a, -2a)$ — its length is $4a$.

For ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: latus rectum endpoints are $(c, b^2/a)$ and $(c, -b^2/a)$ — length = $2b^2/a$.

Director Circle:

For the circle $x^2 + y^2 = a^2$, the director circle doesn’t exist in the same sense. For the ellipse and hyperbola:

• Director circle of ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $x^2 + y^2 = a^2 + b^2$. The locus of intersection of tangents drawn at right angles.
• Director circle of hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $x^2 + y^2 = a^2 - b^2$ (exists only if $a > b$).

Conjugate Hyperbola:

For hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, its conjugate is $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ (swap $a^2$ and $b^2$ terms).

Rectangular (Equilateral) Hyperbola:

When $a = b$ in the standard hyperbola equation: $x^2 - y^2 = a^2$. Its asymptotes are $y = \pm x$ (perpendicular). Its eccentricity is $\sqrt{2}$.

The rectangular hyperbola $xy = c^2$ can be obtained by rotating the standard form by $45°$.

Auxiliary Circle and Eccentric Angle:

For ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, the auxiliary circle is $x^2 + y^2 = a^2$ (the circle with the same major axis as diameter). The eccentric angle $\theta$ relates to position on the ellipse: $x = a\cos\theta$, $y = b\sin\theta$.

Pedal Equations:

• Circle $r = a$: pedal is a cardioid for pole at the circumference
• These are generally beyond ECAT scope but illustrate the depth of conic geometry

Tracing a Conic — Key Points:

For $y^2 = 4ax$ (parabola opening right): vertex at $(0,0)$, passes through $(a, 2a)$, symmetric about x-axis.

For ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$: $a = 5$, $b = 4$, $c = \sqrt{25-16} = 3$. Vertices at $(\pm 5, 0)$ and $(0, \pm 4)$. Foci at $(\pm 3, 0)$. Length of latus rectum = $2 \times 16/5 = 6.4$.

ECAT Previous Year Patterns:

• Tangent equations: very common
• Standard forms and parameters: common
• Condition for tangency: common
• Focus and directrix: occasional
• Eccentricity calculations: periodic

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