JEE Conic Sections Cheatsheet

Cheatsheet Content

### Universal Transformations For a general second-degree equation $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$, the standard replacements for finding $T$ (line equation) are: - $x^2 \rightarrow xx_1$ - $y^2 \rightarrow yy_1$ - $xy \rightarrow \frac{1}{2}(xy_1 + yx_1)$ - $x \rightarrow \frac{1}{2}(x+x_1)$ - $y \rightarrow \frac{1}{2}(y+y_1)$ - $c \rightarrow c$ ### The "Big Four" Universal Equations These formulas apply to **all conics** (Circle, Parabola, Ellipse, Hyperbola) without exception. | Equation | Description | |----------|-------------| | $T=0$ | **Tangent** at $(x_1, y_1)$ on the conic, OR **Chord of Contact** from an external point $(x_1, y_1)$. | | $T=S_1$ | **Chord with Midpoint** $(x_1, y_1)$. ($S_1 = ax_1^2 + 2hx_1y_1 + by_1^2 + 2gx_1 + 2fy_1 + c$) | | $SS_1=T^2$ | **Pair of Tangents** from an external point $(x_1, y_1)$. ($S$ is the conic equation, $S_1$ is $S$ with $(x,y)$ replaced by $(x_1, y_1)$) | ### Tangency & Slope Form ($y = mx + c$) **Universal Shortcut:** To get Hyperbola formulas from Ellipse formulas, replace $b^2 \rightarrow -b^2$. | Conic | Standard Equation | Tangency Condition ($D=0$) | Equation of Tangent | |---|---|---|---| | Parabola | $y^2 = 4ax$ | $c = a/m$ | $y = mx + a/m$ | | Ellipse | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | $c^2 = a^2m^2 + b^2$ | $y = mx \pm \sqrt{a^2m^2 + b^2}$ | | Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | $c^2 = a^2m^2 - b^2$ | $y = mx \pm \sqrt{a^2m^2 - b^2}$ | ### Locus Analogies #### Director Circle The locus of the point of intersection of **perpendicular tangents** to a conic. | Conic | Standard Equation | Director Circle Equation | |---|---|---| | Circle | $x^2 + y^2 = r^2$ | $x^2 + y^2 = 2r^2$ (Concentric circle with radius $\sqrt{2}r$) | | Parabola | $y^2 = 4ax$ | $x = -a$ (**This is the Directrix**) | | Ellipse | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | $x^2 + y^2 = a^2 + b^2$ | | Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | $x^2 + y^2 = a^2 - b^2$ (Real if $a^2 > b^2$, Point if $a^2 = b^2$, Imaginary if $a^2 ### Normals Equations of Normals at a point $(x_1, y_1)$ on the conic. **Sign Symmetries:** Notice the change in sign for $b^2$ between Ellipse and Hyperbola. | Conic | Standard Equation | Equation of Normal | |---|---|---| | Parabola | $y^2 = 4ax$ | $y - y_1 = -\frac{y_1}{2a}(x - x_1)$ | | Ellipse | $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | $\frac{a^2x}{x_1} - \frac{b^2y}{y_1} = a^2 - b^2$ | | Hyperbola | $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ | $\frac{a^2x}{x_1} + \frac{b^2y}{y_1} = a^2 + b^2$ | ### Parametric Coordinates "Bridges" to simplify calculations and trigonometric identities used. | Conic | Parametric Coordinates $(x, y)$ | Trigonometric Identity/Notes | |---|---|---| | Circle | $(r \cos\theta, r \sin\theta)$ | $\cos^2\theta + \sin^2\theta = 1$ | | Parabola | $(at^2, 2at)$ | No trig identity; $t$ is a real parameter. | | Ellipse | $(a \cos\theta, b \sin\theta)$ | $\cos^2\theta + \sin^2\theta = 1$ | | Hyperbola | $(a \sec\theta, b \tan\theta)$ | $\sec^2\theta - \tan^2\theta = 1$ | | Rectangular Hyperbola ($xy=c^2$) | $(ct, c/t)$ | No trig identity; $t$ is a real parameter. |