JEE Advanced Coordinate Geometry

Cheatsheet Content

1. Straight Lines Distance Formula: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$ Section Formula: Internal: $(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n})$ External: $(\frac{m x_2 - n x_1}{m-n}, \frac{m y_2 - n y_1}{m-n})$ Area of Triangle: $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$ Slope of a Line ($m$): Through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2-y_1}{x_2-x_1}$ Angle $\theta$ with positive x-axis: $m = \tan\theta$ Equation of a Line: Slope-Intercept: $y = mx+c$ Point-Slope: $y-y_1 = m(x-x_1)$ Two-Point: $y-y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1)$ Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$ Normal Form: $x\cos\alpha + y\sin\alpha = p$ General Form: $Ax+By+C=0$ Angle between two lines: $\tan\theta = |\frac{m_1-m_2}{1+m_1m_2}|$ Condition for Parallel Lines: $m_1 = m_2$ Condition for Perpendicular Lines: $m_1m_2 = -1$ Distance from a Point to a Line: $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$ Distance between Parallel Lines: $d = \frac{|C_1-C_2|}{\sqrt{A^2+B^2}}$ (for $Ax+By+C_1=0$ and $Ax+By+C_2=0$) Family of Lines: $L_1 + \lambda L_2 = 0$ (Passes through intersection of $L_1=0$ and $L_2=0$) Homogenization: To find chord of contact or tangent from external point. Concurrency of three lines: $A_1x+B_1y+C_1=0$, $A_2x+B_2y+C_2=0$, $A_3x+B_3y+C_3=0$ are concurrent if $\begin{vmatrix} A_1 & B_1 & C_1 \\ A_2 & B_2 & C_2 \\ A_3 & B_3 & C_3 \end{vmatrix} = 0$ 2. Circles General Equation: $x^2+y^2+2gx+2fy+c=0$ Center: $(-g, -f)$ Radius: $r = \sqrt{g^2+f^2-c}$ Standard Equation: $(x-h)^2+(y-k)^2=r^2$ Center: $(h, k)$ Radius: $r$ Parametric Form: $x = h+r\cos\theta$, $y = k+r\sin\theta$ Equation of Tangent: At $(x_1, y_1)$: $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$ Slope $m$: $y=mx \pm r\sqrt{1+m^2}$ (for $x^2+y^2=r^2$) Slope $m$: $y-k=m(x-h) \pm r\sqrt{1+m^2}$ (for $(x-h)^2+(y-k)^2=r^2$) Length of Tangent from $(x_1, y_1)$: $L = \sqrt{x_1^2+y_1^2+2gx_1+2fy_1+c} = \sqrt{S_1}$ Chord of Contact from $(x_1, y_1)$: $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$ Pair of Tangents from $(x_1, y_1)$: $SS_1=T^2$ Director Circle: $x^2+y^2=2r^2$ (Locus of points from which perpendicular tangents can be drawn) Radical Axis of two circles $S_1=0$ and $S_2=0$: $S_1-S_2=0$ Radical Centre: Point of concurrency of radical axes of three circles. Coaxial System of Circles: $S+\lambda L = 0$ (if $L$ is common chord), $S+\lambda S' = 0$ (if $S$ and $S'$ intersect) 3. Parabola Standard Equation: $y^2=4ax$ Vertex: $(0,0)$ Focus: $(a,0)$ Directrix: $x=-a$ Axis: $y=0$ Latus Rectum Length: $4a$ Other Forms: $y^2=-4ax$, $x^2=4ay$, $x^2=-4ay$ Parametric Form: $(at^2, 2at)$ Equation of Tangent: At $(x_1, y_1)$: $yy_1=2a(x+x_1)$ Parametric $t$: $yt=x+at^2$ Slope $m$: $y=mx+\frac{a}{m}$ Normal: At $(x_1, y_1)$: $y-y_1 = -\frac{y_1}{2a}(x-x_1)$ Parametric $t$: $y+tx=2at+at^3$ Slope $m$: $y=mx-2am-am^3$ Chord of Contact from $(x_1, y_1)$: $yy_1=2a(x+x_1)$ Director Circle: $x=-a$ (the directrix itself) 4. Ellipse Standard Equation: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b$) Center: $(0,0)$ Major Axis Length: $2a$ (along x-axis) Minor Axis Length: $2b$ (along y-axis) Foci: $(\pm ae, 0)$ Directrices: $x = \pm \frac{a}{e}$ Eccentricity: $e = \sqrt{1-\frac{b^2}{a^2}}$ Latus Rectum Length: $\frac{2b^2}{a}$ Vertices: $(\pm a, 0)$ Co-vertices: $(0, \pm b)$ Other Form: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (where $a>b$, major axis along y-axis) Foci: $(0, \pm ae)$ Directrices: $y = \pm \frac{a}{e}$ Parametric Form: $(a\cos\theta, b\sin\theta)$ Equation of Tangent: At $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ Parametric $\theta$: $\frac{x\cos\theta}{a} + \frac{y\sin\theta}{b} = 1$ Slope $m$: $y=mx \pm \sqrt{a^2m^2+b^2}$ Chord of Contact from $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ Director Circle: $x^2+y^2=a^2+b^2$ 5. Hyperbola Standard Equation: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Center: $(0,0)$ Transverse Axis Length: $2a$ (along x-axis) Conjugate Axis Length: $2b$ (along y-axis) Foci: $(\pm ae, 0)$ Directrices: $x = \pm \frac{a}{e}$ Eccentricity: $e = \sqrt{1+\frac{b^2}{a^2}}$ Latus Rectum Length: $\frac{2b^2}{a}$ Vertices: $(\pm a, 0)$ Asymptotes: $\frac{x}{a} \pm \frac{y}{b} = 0$ or $y=\pm \frac{b}{a}x$ Conjugate Hyperbola: $\frac{y^2}{b^2} - \frac{x^2}{a^2} = 1$ or $\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1$ Foci: $(0, \pm be')$ (where $e'=\sqrt{1+\frac{a^2}{b^2}}$) Directrices: $y = \pm \frac{b}{e'}$ Rectangular Hyperbola: $xy=c^2$ (asymptotes are axes) Vertices: $(\pm c, \pm c)$ Foci: $(\pm c\sqrt{2}, \pm c\sqrt{2})$ Eccentricity: $e=\sqrt{2}$ Parametric Form: $(a\sec\theta, b\tan\theta)$ (for $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$) Parametric Form for Rectangular Hyperbola: $(ct, c/t)$ Equation of Tangent: At $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ Parametric $\theta$: $\frac{x\sec\theta}{a} - \frac{y\tan\theta}{b} = 1$ Slope $m$: $y=mx \pm \sqrt{a^2m^2-b^2}$ (condition $a^2m^2 > b^2$) Chord of Contact from $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ Director Circle: $x^2+y^2=a^2-b^2$ (real if $a>b$, point if $a=b$, imaginary if $a 6. General Conic Section ($Ax^2+Bxy+Cy^2+Dx+Ey+F=0$) Discriminant: $\Delta = abc+2fgh-af^2-bg^2-ch^2 = \begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}$ (for $ax^2+2hxy+by^2+2gx+2fy+c=0$) Nature of Conic: If $\Delta \ne 0$: $h^2-ab $h^2-ab = 0$: Parabola $h^2-ab > 0$: Hyperbola If $\Delta = 0$: (Degenerate Conic) $h^2-ab $h^2-ab = 0$: Pair of Parallel Lines $h^2-ab > 0$: Pair of Intersecting Lines