math_coordinate_geometryCoordinate geometry
 Distance formulaL=((x1-x2))2+((y1-y2))2$L=\sqrt{\left({x}_{1}-{x}_{2}{\right)}^{2}+\left({y}_{1}-{y}_{2}{\right)}^{2}}$ Slope of a line when coordinates of any two points on the line are givenslope=m=y2-y1x2-x1$slope=m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ are parallel if m1=m2${m}_{1}={m}_{2}$ Two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ are perpendicular if m1.m2=-1${m}_{1}.{m}_{2}=-1$ Angle between two lines with slope m1 and m2${m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}$ tan𝜃=m2-m11+m1m2$tan𝜃=\frac{{m}_{2}-{m}_{1}}{1+{m}_{1}{m}_{2}}$ Equation of straight line: Point slope formulam=y-y0x-x0$m=\frac{y-{y}_{0}}{x-{x}_{0}}$ Equation of straight line: Point slope formulam=y-y0x-x0$m=\frac{y-{y}_{0}}{x-{x}_{0}}$ Equation of straight line: two point formulay-y1x-x1=y2-y1x2-x1$\frac{y-{y}_{1}}{x-{x}_{1}}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ Equation of straight line: slope intercept formy=mx+c$y=mx+c$where c is y-intercept Equation of straight line: intercept formxa+yb=1$\frac{x}{a}+\frac{y}{b}=1$where, a is x-intercept and b is y-intercept Equation of straight line: normal formx cos𝜔+y sin 𝜔=p $x\phantom{\rule{0.22em}{0ex}}cos𝜔+y\phantom{\rule{0.22em}{0ex}}sin\phantom{\rule{0.22em}{0ex}}𝜔=p\phantom{\rule{0.22em}{0ex}}$where, p is normal distance of line from origin and 𝜔 $𝜔\phantom{\rule{0.22em}{0ex}}$ is angle which normal makes with positive x-axis. Conic Section Axis: A line about which conic section is symmetric is called axis of conic section Vertex: The point of intersection of conic section with its axis of symmetry is called vertex. Focal distance: The distance of a point on a conic section from focus is called focal distance of a point. Focal chord: A chord of a conic section passing through its focus is called a focal chord.
Pair of straight lines $\begin{array}{l}Theorem-1:\phantom{\rule{0.22em}{0ex}}The\phantom{\rule{0.22em}{0ex}}joint\phantom{\rule{0.22em}{0ex}}equation\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}pair\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}straight\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}passing\phantom{\rule{0.22em}{0ex}}thru\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}origin\phantom{\rule{0.22em}{0ex}}is\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}\\ homogeneous\phantom{\rule{0.22em}{0ex}}equation\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}degree\phantom{\rule{0.22em}{0ex}}2\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}y\\ \\ Theorem-2:\phantom{\rule{0.22em}{0ex}}Homogeneous\phantom{\rule{0.22em}{0ex}}equation\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}degree\phantom{\rule{0.22em}{0ex}}two\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}y\phantom{\rule{0.22em}{0ex}}.i.e\\ a{x}^{2}+2hxy+b{y}^{2}\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}0\phantom{\rule{0.22em}{0ex}}represents\phantom{\rule{0.22em}{0ex}}a\phantom{\rule{0.22em}{0ex}}pair\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}straight\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}passing\phantom{\rule{0.22em}{0ex}}thru\phantom{\rule{0.22em}{0ex}}origin\phantom{\rule{0.22em}{0ex}}if\phantom{\rule{0.22em}{0ex}}\\ {h}^{2}-ab\phantom{\rule{0.22em}{0ex}}⩾0\\ \\ if\phantom{\rule{0.22em}{0ex}}{h}^{2}-ab\phantom{\rule{0.22em}{0ex}}⩾0\phantom{\rule{0.22em}{0ex}}then\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}are\phantom{\rule{0.22em}{0ex}}real\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}distinct\\ if\phantom{\rule{0.22em}{0ex}}{h}^{2}-ab\phantom{\rule{0.22em}{0ex}}=0\phantom{\rule{0.22em}{0ex}}then\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}are\phantom{\rule{0.22em}{0ex}}real\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}coincident\\ if\phantom{\rule{0.22em}{0ex}}{h}^{2}-ab\phantom{\rule{0.22em}{0ex}}⩽0\phantom{\rule{0.22em}{0ex}}then\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}slope\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}the\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}are\phantom{\rule{0.22em}{0ex}}not\phantom{\rule{0.22em}{0ex}}real\\ \\ if\phantom{\rule{0.22em}{0ex}}{m}_{1}\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}{m}_{2}\phantom{\rule{0.22em}{0ex}}are\phantom{\rule{0.22em}{0ex}}slope\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}then\phantom{\rule{0.22em}{0ex}}\\ {m}_{1}+{m}_{2}=-\frac{2h}{b}\\ {m}_{1}{m}_{2}=\frac{a}{b}\\ \\ General\phantom{\rule{0.22em}{0ex}}second\phantom{\rule{0.22em}{0ex}}degree\phantom{\rule{0.22em}{0ex}}equation\phantom{\rule{0.22em}{0ex}}in\phantom{\rule{0.22em}{0ex}}x\phantom{\rule{0.22em}{0ex}}and\phantom{\rule{0.22em}{0ex}}y\\ \\ a{x}^{2}+2hxy+b{y}^{2}+2gx+2fy+c\phantom{\rule{0.22em}{0ex}}=\phantom{\rule{0.22em}{0ex}}0\phantom{\rule{0.22em}{0ex}}.\phantom{\rule{0.22em}{0ex}}\\ It\phantom{\rule{0.22em}{0ex}}represents\phantom{\rule{0.22em}{0ex}}pair\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}straight\phantom{\rule{0.22em}{0ex}}lines\phantom{\rule{0.22em}{0ex}}only\phantom{\rule{0.22em}{0ex}}if\phantom{\rule{0.22em}{0ex}}\\ \\ |\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}|=0\phantom{\rule{0.22em}{0ex}},\phantom{\rule{0.22em}{0ex}}i.e.\phantom{\rule{0.44em}{0ex}}abc+2fgh-a{f}^{2}-b{g}^{2}-c{h}^{2}=0\\ \\ point\phantom{\rule{0.22em}{0ex}}of\phantom{\rule{0.22em}{0ex}}intersection\phantom{\rule{0.22em}{0ex}}is:\phantom{\rule{0.22em}{0ex}}\left(\frac{hf-bg}{ab-{h}^{2}},\frac{gh-af}{ab-{h}^{2}}\right)\\ \\ \\ \\ \\ \\ \end{array}$