“Analytical Geometry: Lines and Conics”

🟢 Lite — Quick Review (1h–1d)

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

Analytical geometry (coordinate geometry) combines algebra and geometry — equations represent geometric shapes, and geometric concepts solve algebraic problems.

Equation of a Straight Line:

Form	Equation	Use
Slope-intercept	$y = mx + c$	$m$ = slope, $c$ = y-intercept
Point-slope	$y - y_1 = m(x - x_1)$	Through $(x_1, y_1)$ with slope $m$
Two-point	$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$	Through $(x_1,y_1)$ and $(x_2,y_2)$
Intercept form	$\frac{x}{a} + \frac{y}{b} = 1$	x-intercept $a$, y-intercept $b$
General form	$Ax + By + C = 0$	General form, $A^2 + B^2 \neq 0$

Key Relationships:

• Slope $m = \tan \theta = \frac{y_2 - y_1}{x_2 - x_1}$
• Angle between two lines: $\tan \theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|$
• Perpendicular: $m_1 m_2 = -1$
• Parallel: $m_1 = m_2$
• Distance from point $(x_1,y_1)$ to line $Ax + By + C = 0$: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

⚡ ECAT exam tips:

• For a line making equal intercepts on axes: $x + y = a$ (not $x + y = c$ with different coefficients)
• For a line passing through origin: $C = 0$
• For a line parallel to $x$-axis: $y = c$ (slope = 0)
• For a line parallel to $y$-axis: $x = c$ (slope undefined)

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🟡 Standard — Regular Study (2d–2mo)

For ECAT students who want genuine understanding of coordinate geometry.

Conic Sections — The Four Curves:

Conic sections are curves obtained by intersecting a plane with a double cone. The four standard conics:

1. Parabola: $y^2 = 4ax$ (opens right), $x^2 = 4ay$ (opens up)

• Focus: $(a, 0)$, Directrix: $x = -a$
• Focal parameter (latus rectum): $4a$
• Any point $(at^2, 2at)$; parametric form

2. 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 $\sqrt{g^2 + f^2 - c}$

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

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

4. Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (opens left-right)

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

Standard Results for Circles:

• Equation of tangent to circle $x^2 + y^2 = a^2$ at $(x_1, y_1)$: $xx_1 + yy_1 = a^2$
• Length of tangent from external point $(x_1,y_1)$ to $x^2 + y^2 = a^2$: $\sqrt{x_1^2 + y_1^2 - a^2}$
• Condition for $y = mx + c$ to be tangent to $x^2 + y^2 = a^2$: $c^2 = a^2(1 + m^2)$

Standard Results for Parabola $y^2 = 4ax$:

• Tangent at $(at_1^2, 2at_1)$: $ty_1 = x + at_1^2$ → $ty = x + at^2$
• Condition for $y = mx + c$ to be tangent: $c = a/m$
• Point of intersection of tangents at $t_1$ and $t_2$: $(at_1 t_2, a(t_1 + t_2))$

⚡ Common student mistakes:

1. Forgetting that the general form of a circle $x^2 + y^2 + 2gx + 2fy + c = 0$ has centre $(-g, -f)$, not $(g, f)$
2. Confusing $a^2 - b^2$ (ellipse: $c^2 = a^2 - b^2$) with $a^2 + b^2$ (hyperbola: $c^2 = a^2 + b^2$)
3. For parabola $y^2 = 4ax$: the focus is at $(a, 0)$ — not at $(0, a)$
4. Using the wrong formula for distance from point to line — must be absolute value

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for ECAT mastery of analytical geometry.

Transformation of Axes:

• Shift of origin (without rotation): $x = X + h$, $y = Y + k$
• Rotation of axes by angle $\theta$: $x = X \cos\theta - Y \sin\theta$, $y = X \sin\theta + Y \cos\theta$

When axes are rotated by $\theta$ to eliminate the $xy$ term in $Ax^2 + Bxy + Cy^2 + … = 0$: $$\tan 2\theta = \frac{B}{A - C}$$

Pair of Straight Lines:

The homogeneous equation $ax^2 + 2hxy + by^2 = 0$ represents two straight lines through the origin. The lines are real and distinct if $h^2 > ab$, coincident if $h^2 = ab$, and imaginary if $h^2 < ab$.

The angle between them: $\tan \theta = \frac{2\sqrt{h^2 - ab}}{a + b}$.

For non-homogeneous pair $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, the lines pass through the point of intersection of $ax^2 + 2hxy + by^2 = 0$ and $gx + fy + c = 0$.

Joint Equation of Two Lines:

The pair of lines joining the origin to the intersection points of two curves $S = 0$ and $S’ = 0$ is given by $S + \lambda S’ = 0$ (homogeneous case) or more generally by the eliminant.

Important Formulae for Lines:

• Perpendicular distance between two parallel lines $y = mx + c_1$ and $y = mx + c_2$: $d = \frac{|c_2 - c_1|}{\sqrt{1+m^2}}$
• Area of triangle with vertices $(x_1,y_1), (x_2,y_2), (x_3,y_3)$: $\Delta = \frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$
• Coordinates of centroid: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)$
• Coordinates of midpoint of $(x_1,y_1)$ and $(x_2,y_2)$: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

Ellipse — Detailed Properties:

Parametric form: $x = a \cos\theta$, $y = b \sin\theta$. Equation of tangent at $(x_1,y_1)$ on $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$. The sum of distances from any point on the ellipse to the two foci is constant = $2a$.

Hyperbola — Detailed Properties:

Rectangular hyperbola: $xy = c^2$ or equivalently $x^2 - y^2 = a^2$ rotated by $45°$. Asymptotes of standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $y = \pm \frac{b}{a}x$. The difference of distances from any point on the hyperbola to the two foci is constant = $2a$.

Locus Problems:

A locus is the set of all points satisfying a given condition.

Example: Find the locus of a point $P$ such that its distances from $(1,2)$ and $(-1,3)$ are equal. $\sqrt{(x-1)^2 + (y-2)^2} = \sqrt{(x+1)^2 + (y-3)^2}$ Squaring and simplifying: $x^2 - 2x + 1 + y^2 - 4y + 4 = x^2 + 2x + 1 + y^2 - 6y + 9$ Cancelling $x^2 + y^2 + 1$: $-2x - 4y + 4 = 2x - 6y + 9$ $-4x + 2y - 5 = 0$ or $4x - 2y + 5 = 0$ — a straight line.

ECAT Previous Year Patterns:

• Equation of line and conditions: very common
• Distance from point to line: calculation-based, common
• Circle equations and tangents: very common
• Parabola tangents: common
• Locus problems: periodically tested
• Angle between lines: occasional

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