Coordinate Geometry JEE

Cheatsheet Content

### Basics of Coordinate Geometry - **Distance Formula:** Distance between $P(x_1, y_1)$ and $Q(x_2, y_2)$ is $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **Section Formula:** - **Internal Division:** Point $(x,y)$ dividing $PQ$ in ratio $m:n$ is $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$. - **External Division:** Point $(x,y)$ dividing $PQ$ in ratio $m:n$ is $\left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}\right)$. - **Midpoint Formula:** Midpoint of $PQ$ is $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. - **Area of a Triangle:** For vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3)$, area is $\frac{1}{2} |x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)|$. - **Collinearity of Three Points:** Points are collinear if the area of the triangle formed by them is zero. ### Straight Lines - **Slope of a Line (m):** - Passing through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2-y_1}{x_2-x_1}$. - If line is $ax+by+c=0$: $m = -\frac{a}{b}$. - Angle with positive x-axis $\theta$: $m = \tan\theta$. - **Equations of a Line:** - **Slope-intercept form:** $y = mx + c$ - **Point-slope form:** $y - y_1 = m(x - x_1)$ - **Two-point form:** $y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x - x_1)$ - **Intercept form:** $\frac{x}{a} + \frac{y}{b} = 1$ - **Normal form:** $x\cos\alpha + y\sin\alpha = p$ (p = perpendicular distance from origin) - **Parametric form:** $\frac{x-x_1}{\cos\theta} = \frac{y-y_1}{\sin\theta} = r$ - **Angle Between Two Lines:** If slopes are $m_1, m_2$, then $\tan\theta = \left|\frac{m_1-m_2}{1+m_1m_2}\right|$. - Parallel lines: $m_1 = m_2$. - Perpendicular lines: $m_1m_2 = -1$. - **Distance of a Point from a Line:** Distance from $(x_1, y_1)$ to $ax+by+c=0$ is $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$. - **Distance Between Parallel Lines:** For $ax+by+c_1=0$ and $ax+by+c_2=0$, distance is $\frac{|c_1-c_2|}{\sqrt{a^2+b^2}}$. ### Pair of Straight Lines - **Homogeneous Equation of Second Degree:** $ax^2 + 2hxy + by^2 = 0$ represents two lines passing through the origin. - If $h^2 > ab$, lines are real and distinct. - If $h^2 = ab$, lines are real and coincident. - If $h^2 ### Circle - **Standard Equation:** $(x-h)^2 + (y-k)^2 = r^2$, center $(h,k)$, radius $r$. - **General Equation:** $x^2+y^2+2gx+2fy+c=0$, center $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$. - **Equation of Tangent:** - At $(x_1, y_1)$ on $x^2+y^2+2gx+2fy+c=0$: $xx_1+yy_1+g(x+x_1)+f(y+y_1)+c=0$. - Slope $m$ to $x^2+y^2=a^2$: $y = mx \pm a\sqrt{1+m^2}$. - **Length of Tangent:** From $(x_1, y_1)$ to $S=0$: $\sqrt{S_1}$, where $S_1 = x_1^2+y_1^2+2gx_1+2fy_1+c$. - **Power of a Point:** For a point $P(x_1, y_1)$ and a circle $S=0$, the power of the point is $S_1$. - **Radical Axis:** For two circles $S_1=0$ and $S_2=0$, the radical axis is $S_1-S_2=0$. ### Parabola - **Standard Equation:** $y^2 = 4ax$ - Vertex: $(0,0)$ - Focus: $(a,0)$ - Directrix: $x=-a$ - Latus Rectum length: $4a$ - **Parametric Equations:** $(at^2, 2at)$ - **Equation of Tangent:** - At $(x_1, y_1)$: $yy_1 = 2a(x+x_1)$ - Slope $m$: $y = mx + \frac{a}{m}$ - **Condition for Tangency:** Line $y=mx+c$ is tangent to $y^2=4ax$ if $c = a/m$. ### Ellipse - **Standard Equation:** $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a>b$) - Center: $(0,0)$ - Vertices: $(\pm a, 0)$ - Foci: $(\pm ae, 0)$ - Directrices: $x = \pm a/e$ - Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ - Latus Rectum length: $\frac{2b^2}{a}$ - **Parametric Equations:** $(a\cos\theta, b\sin\theta)$ - **Equation of Tangent:** - At $(x_1, y_1)$: $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$ - Slope $m$: $y = mx \pm \sqrt{a^2m^2+b^2}$ - **Condition for Tangency:** Line $y=mx+c$ is tangent to $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ if $c^2 = a^2m^2+b^2$. ### Hyperbola - **Standard Equation:** $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ - Center: $(0,0)$ - Vertices: $(\pm a, 0)$ - Foci: $(\pm ae, 0)$ - Directrices: $x = \pm a/e$ - Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$ - Latus Rectum length: $\frac{2b^2}{a}$ - Asymptotes: $\frac{x}{a} \pm \frac{y}{b} = 0$ - **Parametric Equations:** $(a\sec\theta, b\tan\theta)$ - **Equation of Tangent:** - At $(x_1, y_1)$: $\frac{xx_1}{a^2} - \frac{yy_1}{b^2} = 1$ - Slope $m$: $y = mx \pm \sqrt{a^2m^2-b^2}$ - **Condition for Tangency:** Line $y=mx+c$ is tangent to $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ if $c^2 = a^2m^2-b^2$. ### General Conic Section - **Definition:** Locus of a point whose distance from a fixed point (focus) is a constant ratio ($e$, eccentricity) to its distance from a fixed line (directrix). - **Equation:** $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ - **Nature of Conic:** Determined by the discriminant $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2$ and $h^2-ab$. - If $\Delta \neq 0$: - $h^2-ab = 0$: Parabola - $h^2-ab 0$: Hyperbola - If $\Delta = 0$: Pair of straight lines (degenerate conic).