Coordinate Geometry Formula Sheet

A single scannable sheet of every must-know Coordinate Geometry formula - basics, straight lines, circles, and all four conics - for last-minute JEE Main and Advanced revision. Group by sub-topic and skim the headline boxed formulas first.

Basics: Points, Distances, Division

Quantity	Formula	Notes
Distance between two points	$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$	Pythagoras
Section (internal, $m:n$)	$\left(\dfrac{mx_2 + nx_1}{m+n}, \dfrac{my_2 + ny_1}{m+n}\right)$	Point divides internally
Section (external, $m:n$)	$\left(\dfrac{mx_2 - nx_1}{m-n}, \dfrac{my_2 - ny_1}{m-n}\right)$	Point divides externally
Midpoint	$\left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$	$m:n = 1:1$
Centroid of triangle	$\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)$	Average of three vertices

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Straight Lines

Forms of the Line

Form	Equation
Slope-intercept	$y = mx + c$
Point-slope	$y - y_1 = m(x - x_1)$
Two-point	$\dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1}$
Intercept	$\dfrac{x}{a} + \dfrac{y}{b} = 1$
Normal (perpendicular)	$x\cos\alpha + y\sin\alpha = p$
General	$Ax + By + C = 0$

For the general form $Ax + By + C = 0$: slope $m = -\dfrac{A}{B}$, x-intercept $= -\dfrac{C}{A}$, y-intercept $= -\dfrac{C}{B}$.

Slope and Angle

$$\boxed{m = \tan\theta = \frac{y_2 - y_1}{x_2 - x_1}}$$$$\boxed{\tan\theta = \left|\frac{m_1 - m_2}{1 + m_1 m_2}\right|}$$

Relation	Slope condition	Coefficient condition
Parallel	$m_1 = m_2$	$\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2} \neq \dfrac{C_1}{C_2}$
Perpendicular	$m_1 m_2 = -1$	$A_1 A_2 + B_1 B_2 = 0$
Coincident	same line	$\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2} = \dfrac{C_1}{C_2}$

Distances

$$\boxed{d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}} \qquad \text{(point to line)}$$$$\boxed{d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}} \qquad \text{(between parallel lines, same } A,B)$$

Other Key Results

Quantity	Formula	Notes
Foot of perpendicular from $(x_1,y_1)$ to $ax+by+c=0$	$\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = -\dfrac{ax_1 + by_1 + c}{a^2 + b^2}$	Gives foot directly
Family of lines through intersection	$L_1 + \lambda L_2 = 0$	Avoids solving for intersection
Triangle area from intercept form $\frac{x}{a}+\frac{y}{b}=1$	$\dfrac{1}{2}\lvert ab\rvert$	With coordinate axes

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Circles

Equations

$$\boxed{(x - h)^2 + (y - k)^2 = r^2} \qquad \text{center } (h,k),\ \text{radius } r$$$$\boxed{x^2 + y^2 + 2gx + 2fy + c = 0}$$

Quantity	From general form
Center	$(-g, -f)$
Radius	$\sqrt{g^2 + f^2 - c}$
Real circle	$g^2 + f^2 - c > 0$ (point circle if $=0$, imaginary if $<0$)
Diameter-endpoint form	$(x-x_1)(x-x_2) + (y-y_1)(y-y_2) = 0$
Parametric	$x = h + r\cos\theta,\ y = k + r\sin\theta$

Tangent, Normal, Chords

Result	Formula
Tangent at $(x_1,y_1)$ on $x^2+y^2=r^2$	$xx_1 + yy_1 = r^2$
Tangent at $(x_1,y_1)$ on general circle	$xx_1 + yy_1 + g(x+x_1) + f(y+y_1) + c = 0$
Tangent with slope $m$ on $x^2+y^2=r^2$	$y = mx \pm r\sqrt{1 + m^2}$
Tangency condition for $y=mx+c$	$c^2 = r^2(1 + m^2)$
Normal	always passes through the center (use two-point form)
Length of tangent from $(x_1,y_1)$	$\sqrt{x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c}$
Chord length (perp. distance $d$)	$2\sqrt{r^2 - d^2}$
Chord of contact from $(x_1,y_1)$	$xx_1 + yy_1 = r^2$

Position of a Point and Families

With $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$: $S_1>0$ outside, $S_1=0$ on, $S_1<0$ inside.

$$\boxed{S_1 + \lambda S_2 = 0 \ \text{(family through intersection)}, \qquad S_1 - S_2 = 0 \ \text{(radical axis / common chord)}}$$

Concentric circles: keep $g, f$, vary the constant. Orthogonal circles condition:

$$\boxed{2g_1 g_2 + 2f_1 f_2 = c_1 + c_2}$$

Director circle of $x^2+y^2=a^2$ (locus of perpendicular tangents): $x^2 + y^2 = 2a^2$.

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Parabola

Standard Forms

Equation	Vertex	Focus	Directrix	Axis	Opens	Latus rectum
$y^2 = 4ax$	$(0,0)$	$(a,0)$	$x=-a$	$y=0$	Right	$4a$
$y^2 = -4ax$	$(0,0)$	$(-a,0)$	$x=a$	$y=0$	Left	$4a$
$x^2 = 4ay$	$(0,0)$	$(0,a)$	$y=-a$	$x=0$	Up	$4a$
$x^2 = -4ay$	$(0,0)$	$(0,-a)$	$y=a$	$x=0$	Down	$4a$

Eccentricity $e = 1$ for every parabola. LR endpoints of $y^2=4ax$: $(a, 2a)$ and $(a, -2a)$.

Vertex at $(h,k)$: $(y-k)^2 = 4a(x-h)$ -> focus $(h+a, k)$, directrix $x = h-a$.

Parametric, Tangent, Normal

$$\boxed{x = at^2, \quad y = 2at}$$

Result	Formula
Tangent at $(x_1,y_1)$	$yy_1 = 2a(x + x_1)$
Tangent at parameter $t$	$ty = x + at^2$
Tangent with slope $m$	$y = mx + \dfrac{a}{m}$ (point of contact $\left(\tfrac{a}{m^2}, \tfrac{2a}{m}\right)$)
Tangency condition for $y=mx+c$	$c = \dfrac{a}{m}$
Normal at $(x_1,y_1)$	$y - y_1 = -\dfrac{y_1}{2a}(x - x_1)$
Normal at parameter $t$	$y + tx = 2at + at^3$ (slope $-t$)
Normal with slope $m$	$y = mx - 2am - am^3$

Focal Chord and Chords

$$\boxed{t_1 t_2 = -1 \quad \text{(ends of a focal chord)}}$$

Result	Formula
Focal chord length	$a(t_1 - t_2)^2 = a\left(t_1 + \tfrac{1}{t_1}\right)^2$
Harmonic relation of segments $p,q$	$\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{a}$ (semi-LR = HM of segments)
Intersection of tangents at $t_1, t_2$	$\left(a t_1 t_2,\ a(t_1 + t_2)\right)$
Chord with midpoint $(x_1,y_1)$	$yy_1 - 2a(x + x_1) = y_1^2 - 4ax_1$ (i.e. $T = S_1$)
Locus of perpendicular tangents	directrix $x = -a$

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Ellipse: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$

Key Elements (horizontal, $a > b$)

Element	Value
Center	$(0,0)$
Major / minor axis length	$2a$ / $2b$
Vertices	$(\pm a, 0)$
Co-vertices	$(0, \pm b)$
Foci	$(\pm c, 0)$, $c = \sqrt{a^2 - b^2}$
Eccentricity	$e = \dfrac{c}{a} = \dfrac{\sqrt{a^2 - b^2}}{a}$, $0 < e < 1$
Latus rectum	$\dfrac{2b^2}{a}$
Directrices	$x = \pm \dfrac{a}{e}$

Vertical ellipse $\dfrac{x^2}{b^2} + \dfrac{y^2}{a^2} = 1$ ($a>b$): foci $(0, \pm c)$, vertices $(0, \pm a)$. Larger denominator = major axis.

Core Relations

$$\boxed{c^2 = a^2 - b^2} \qquad \boxed{e = \frac{c}{a}} \qquad \boxed{b^2 = a^2(1 - e^2)}$$

Parametric, Tangent, Normal, Properties

$$\boxed{x = a\cos\theta, \quad y = b\sin\theta}$$

Result	Formula
Tangent at $(x_1,y_1)$	$\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1$
Tangent at parameter $\theta$	$\dfrac{x\cos\theta}{a} + \dfrac{y\sin\theta}{b} = 1$
Tangent with slope $m$	$y = mx \pm \sqrt{a^2 m^2 + b^2}$
Tangency condition for $y=mx+c$	$c^2 = a^2 m^2 + b^2$
Normal at $(x_1,y_1)$	$\dfrac{a^2 x}{x_1} - \dfrac{b^2 y}{y_1} = a^2 - b^2$
Normal at parameter $\theta$	$ax\sec\theta - by\csc\theta = a^2 - b^2$
Sum of focal distances	$PF_1 + PF_2 = 2a$
Chord of contact from $(x_1,y_1)$	$\dfrac{xx_1}{a^2} + \dfrac{yy_1}{b^2} = 1$
Auxiliary circle	$x^2 + y^2 = a^2$
Director circle	$x^2 + y^2 = a^2 + b^2$
Max / min focal distance	$a + c$ / $a - c$

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Hyperbola: $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$

Key Elements (horizontal)

Element	Value
Center	$(0,0)$
Transverse / conjugate axis	$2a$ / $2b$
Vertices	$(\pm a, 0)$
Foci	$(\pm c, 0)$, $c = \sqrt{a^2 + b^2}$
Eccentricity	$e = \dfrac{c}{a} = \dfrac{\sqrt{a^2 + b^2}}{a}$, $e > 1$
Latus rectum	$\dfrac{2b^2}{a}$
Asymptotes	$y = \pm \dfrac{b}{a}x$
Directrices	$x = \pm \dfrac{a}{e}$

Vertical hyperbola $\dfrac{y^2}{a^2} - \dfrac{x^2}{b^2} = 1$: foci $(0, \pm c)$, asymptotes $y = \pm \dfrac{a}{b}x$. The positive term gives the opening direction.

Core Relations

$$\boxed{c^2 = a^2 + b^2} \qquad \boxed{e = \frac{c}{a} > 1} \qquad \boxed{b^2 = a^2(e^2 - 1)} \qquad \boxed{c^2 = a^2 e^2}$$

Asymptotes: set the equation to $0$, i.e. $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 0$.

Parametric, Tangent, Normal, Properties

$$\boxed{x = a\sec\theta, \quad y = b\tan\theta}$$

Result	Formula
Tangent at $(x_1,y_1)$	$\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1$
Tangent at parameter $\theta$	$\dfrac{x\sec\theta}{a} - \dfrac{y\tan\theta}{b} = 1$
Tangent with slope $m$	$y = mx \pm \sqrt{a^2 m^2 - b^2}$ (needs $\lvert m\rvert > \tfrac{b}{a}$)
Tangency condition for $y=mx+c$	$c^2 = a^2 m^2 - b^2$
Normal at $(x_1,y_1)$	$\dfrac{a^2 x}{x_1} + \dfrac{b^2 y}{y_1} = a^2 + b^2$
Normal at parameter $\theta$	$ax\cos\theta + by\cot\theta = a^2 + b^2$
Difference of focal distances	$\lvert PF_1 - PF_2\rvert = 2a$
Chord of contact from $(x_1,y_1)$	$\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1$
Director circle	$x^2 + y^2 = a^2 - b^2$ (real only if $a > b$)
Conjugate hyperbola	$\dfrac{y^2}{b^2} - \dfrac{x^2}{a^2} = 1$ (same asymptotes)

Rectangular Hyperbola

$$\boxed{a = b \iff e = \sqrt{2}} \qquad x^2 - y^2 = a^2 \quad\text{or, rotated 45°,}\quad xy = c^2$$

Result (for $xy = c^2$)	Formula
Parametric	$x = ct,\ y = \dfrac{c}{t}$ ($t \neq 0$)
Asymptotes	$x = 0$ and $y = 0$
Tangent at $(ct, c/t)$	$\dfrac{x}{t} + ty = 2c$
Tangent at $(x_1,y_1)$	$xy_1 + yx_1 = 2c^2$
Normal at $(ct, c/t)$	$xt^3 - yt - ct^4 + c = 0$
Chord of contact from $(x_1,y_1)$	$xy_1 + yx_1 = 2c^2$

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General Second-Degree Conic

$$\boxed{ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0}$$$$\boxed{\Delta = abc + 2fgh - af^2 - bg^2 - ch^2}$$

Classification (non-degenerate, $\Delta \neq 0$)

Condition	Conic	Special case
$h^2 - ab < 0$	Ellipse	Circle if $a = b$, $h = 0$
$h^2 - ab = 0$	Parabola	—
$h^2 - ab > 0$	Hyperbola	Rectangular if $a + b = 0$

Degenerate cases ($\Delta = 0$): $h^2-ab<0 \to$ point; $=0 \to$ two coincident lines; $>0 \to$ two intersecting lines.

Centre, Axes, Asymptotes

$$\boxed{ax_0 + hy_0 + g = 0, \quad hx_0 + by_0 + f = 0} \implies x_0 = \frac{hf - bg}{ab - h^2},\ \ y_0 = \frac{gh - af}{ab - h^2}$$

Parabola has no centre ($ab - h^2 = 0$). Principal axes: $\tan 2\theta = \dfrac{2h}{a-b}$ (same angle removes the $xy$ term under rotation). Asymptotes of a central conic (origin at centre): $ax^2 + 2hxy + by^2 = 0$.

Pair of Straight Lines

$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ is a pair of lines iff $\Delta = 0$.

Homogeneous $ax^2 + 2hxy + by^2 = 0$ (lines through origin): real distinct if $h^2 > ab$, coincident if $h^2 = ab$, imaginary if $h^2 < ab$.

$$\boxed{\tan\theta = \frac{2\sqrt{h^2 - ab}}{a + b}} \qquad \text{perpendicular lines} \iff a + b = 0$$

Invariants Under Rotation

$a + b$, $\ ab - h^2$, and $\Delta$ all stay constant.

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Universal Conic Tools (T-rule)

For any conic, apply the T-substitution to its equation to get tangent / chord of contact / polar:

$$\boxed{x^2 \to xx_1,\quad y^2 \to yy_1,\quad xy \to \tfrac{xy_1 + x_1y}{2},\quad x \to \tfrac{x+x_1}{2},\quad y \to \tfrac{y+y_1}{2}}$$

Use case	Equation
Tangent (point on conic)	$T = 0$
Chord of contact / polar of $(x_1,y_1)$	$T = 0$ (same form)
Chord with given midpoint $(h,k)$	$T = S_1$
Conjugate points $P, Q$	$P$ lies on polar of $Q$

Chord with Midpoint $(h,k)$ — by conic

Conic	Chord ($T = S_1$)
Circle $x^2+y^2=r^2$	$hx + ky = h^2 + k^2$
Parabola $y^2=4ax$	$ky - 2a(x+h) = k^2 - 4ah$
Ellipse	$\dfrac{hx}{a^2} + \dfrac{ky}{b^2} = \dfrac{h^2}{a^2} + \dfrac{k^2}{b^2}$
Hyperbola	$\dfrac{hx}{a^2} - \dfrac{ky}{b^2} = \dfrac{h^2}{a^2} - \dfrac{k^2}{b^2}$

Conjugate Points Condition

Conic	Condition
Circle	$x_1 x_2 + y_1 y_2 = r^2$
Parabola	$y_1 y_2 = 2a(x_1 + x_2)$
Ellipse	$\dfrac{x_1 x_2}{a^2} + \dfrac{y_1 y_2}{b^2} = 1$
Hyperbola	$\dfrac{x_1 x_2}{a^2} - \dfrac{y_1 y_2}{b^2} = 1$

Pole of line $hx + ky = r^2$ vs circle $x^2+y^2=r^2$ is $(h,k)$; for line $lx+my+n=0$ vs $x^2+y^2=r^2$ the pole is $\left(\dfrac{r^2 l}{n}, \dfrac{r^2 m}{n}\right)$. Length of chord of contact from $(x_1,y_1)$ to $x^2+y^2=r^2$: $\dfrac{2r\sqrt{x_1^2+y_1^2-r^2}}{\sqrt{x_1^2+y_1^2}}$.

Director Circle (locus of perpendicular tangents)

Conic	Locus
Parabola $y^2 = 4ax$	directrix $x = -a$
Ellipse	$x^2 + y^2 = a^2 + b^2$
Hyperbola	$x^2 + y^2 = a^2 - b^2$ (if $a>b$)
Circle $x^2+y^2=r^2$	$x^2 + y^2 = 2r^2$

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Eccentricity Snapshot

Conic	Eccentricity	Focal-distance property
Circle	$e = 0$	foci coincide at centre
Ellipse	$0 < e < 1$	sum $= 2a$
Parabola	$e = 1$	from focus = from directrix
Hyperbola	$e > 1$	difference $= 2a$
Rectangular hyperbola	$e = \sqrt{2}$	$a = b$

graph TD
    A[General conic: ax^2 + 2hxy + by^2 + ...] --> B{Δ = 0?}
    B -->|Yes| C[Degenerate: point / lines]
    B -->|No| D{h^2 - ab}
    D -->|< 0| E[Ellipse / Circle]
    D -->|= 0| F[Parabola]
    D -->|> 0| G[Hyperbola / Rectangular]