Two Dimensional Analytical Geometry-II — Formula Sheet
Every key result for the circle and the three conics, grouped for quick revision. Throughout, a tangent line in slope form is \(y=mx+c\).
A. Circle
- Standard (centre–radius) form: \((x-h)^{2}+(y-k)^{2}=r^{2}\); centre \((h,k)\), radius \(r\).
- Centred at the origin: \(x^{2}+y^{2}=r^{2}\).
- General form: \(x^{2}+y^{2}+2gx+2fy+c=0\); centre \((-g,-f)\), radius \(\sqrt{g^{2}+f^{2}-c}\).
- Real if \(g^{2}+f^{2}-c>0\); a point if \(=0\); imaginary if \(<0\).
- Diameter form (ends \((x_{1},y_{1}),(x_{2},y_{2})\)): \((x-x_{1})(x-x_{2})+(y-y_{1})(y-y_{2})=0\).
- Family through a line–circle intersection: \(S+\lambda L=0\).
- Position of \((x_{1},y_{1})\): with \(S_{1}=x_{1}^{2}+y_{1}^{2}+2gx_{1}+2fy_{1}+c\), the point is outside / on / inside as \(S_{1}>0/=0/<0\). Length of tangent from it \(=\sqrt{S_{1}}\).
- Tangent at \((x_{1},y_{1})\) on \(x^{2}+y^{2}=a^{2}\): \(xx_{1}+yy_{1}=a^{2}\).
- Tangent at \((x_{1},y_{1})\) on the general circle: \(xx_{1}+yy_{1}+g(x+x_{1})+f(y+y_{1})+c=0\).
- Normal on \(x^{2}+y^{2}=a^{2}\): passes through the centre, \(xy_{1}-yx_{1}=0\).
- Tangency of \(y=mx+c\) to \(x^{2}+y^{2}=a^{2}\): \(c^{2}=a^{2}(1+m^{2})\); tangents \(y=mx\pm a\sqrt{1+m^{2}}\).
B. Eccentricity test
\(e=1\Rightarrow\) parabola, \(01\Rightarrow\) hyperbola, \(e=0\Rightarrow\) circle.
C. Parabola (vertex at origin)
| Equation | Opens | Focus | Directrix | Axis | Latus rectum |
|---|
| \(y^{2}=4ax\) | right | \((a,0)\) | \(x=-a\) | \(x\)-axis | \(4a\) |
| \(y^{2}=-4ax\) | left | \((-a,0)\) | \(x=a\) | \(x\)-axis | \(4a\) |
| \(x^{2}=4ay\) | up | \((0,a)\) | \(y=-a\) | \(y\)-axis | \(4a\) |
| \(x^{2}=-4ay\) | down | \((0,-a)\) | \(y=a\) | \(y\)-axis | \(4a\) |
- Tangent at \((x_{1},y_{1})\) on \(y^{2}=4ax\): \(yy_{1}=2a(x+x_{1})\).
- Tangent at \(t\) (point \((at^{2},2at)\)): \(ty=x+at^{2}\).
- Tangency of \(y=mx+c\): \(c=\dfrac{a}{m}\); point of contact \(\left(\dfrac{a}{m^{2}},\dfrac{2a}{m}\right)\).
- Normal at \((x_{1},y_{1})\): \(y-y_{1}=-\dfrac{y_{1}}{2a}(x-x_{1})\).
- Normal at \(t\): \(y+tx=2at+at^{3}\).
D. Ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) (\(a>b\))
- Vertices \((\pm a,0)\); foci \((\pm c,0)\) with \(c^{2}=a^{2}-b^{2}\); \(e=\dfrac{c}{a}\); directrices \(x=\pm\dfrac{a}{e}\).
- Major axis \(2a\), minor axis \(2b\), latus rectum \(\dfrac{2b^{2}}{a}\); sum of focal distances \(=2a\).
- If \(b>a\): major axis vertical, foci \((0,\pm c)\) with \(c^{2}=b^{2}-a^{2}\).
- Tangent at \((x_{1},y_{1})\): \(\dfrac{xx_{1}}{a^{2}}+\dfrac{yy_{1}}{b^{2}}=1\).
- Tangent at \(\theta\): \(\dfrac{x\cos\theta}{a}+\dfrac{y\sin\theta}{b}=1\).
- Tangency of \(y=mx+c\): \(c^{2}=a^{2}m^{2}+b^{2}\).
- Normal at \((x_{1},y_{1})\): \(\dfrac{a^{2}x}{x_{1}}-\dfrac{b^{2}y}{y_{1}}=a^{2}-b^{2}\).
- Auxiliary circle: \(x^{2}+y^{2}=a^{2}\).
E. Hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\)
- Vertices \((\pm a,0)\); foci \((\pm c,0)\) with \(c^{2}=a^{2}+b^{2}\); \(e=\dfrac{c}{a}>1\); directrices \(x=\pm\dfrac{a}{e}\).
- Transverse axis \(2a\), conjugate axis \(2b\), latus rectum \(\dfrac{2b^{2}}{a}\); \(\bigl|\)difference of focal distances\(\bigr|=2a\).
- Asymptotes: \(y=\pm\dfrac{b}{a}x\).
- Tangent at \((x_{1},y_{1})\): \(\dfrac{xx_{1}}{a^{2}}-\dfrac{yy_{1}}{b^{2}}=1\).
- Tangent at \(\theta\): \(\dfrac{x\sec\theta}{a}-\dfrac{y\tan\theta}{b}=1\).
- Tangency of \(y=mx+c\): \(c^{2}=a^{2}m^{2}-b^{2}\).
- Normal at \((x_{1},y_{1})\): \(\dfrac{a^{2}x}{x_{1}}+\dfrac{b^{2}y}{y_{1}}=a^{2}+b^{2}\).
F. Parametric forms
| Conic | Point | Parameter |
|---|
| Circle \(x^{2}+y^{2}=a^{2}\) | \((a\cos\theta,\,a\sin\theta)\) | \(0\le\theta<2\pi\) |
| Parabola \(y^{2}=4ax\) | \((at^{2},\,2at)\) | \(t\in\mathbb{R}\) |
| Ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) | \((a\cos\theta,\,b\sin\theta)\) | \(0\le\theta<2\pi\) |
| Hyperbola \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) | \((a\sec\theta,\,b\tan\theta)\) | \(\theta\neq\dfrac{\pi}{2},\dfrac{3\pi}{2}\) |
G. Tangency conditions for \(y=mx+c\)
| Conic | Condition | Point of contact |
|---|
| \(x^{2}+y^{2}=a^{2}\) | \(c^{2}=a^{2}(1+m^{2})\) | \(\left(-\dfrac{a^{2}m}{c},\dfrac{a^{2}}{c}\right)\) |
| \(y^{2}=4ax\) | \(c=\dfrac{a}{m}\) | \(\left(\dfrac{a}{m^{2}},\dfrac{2a}{m}\right)\) |
| \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) | \(c^{2}=a^{2}m^{2}+b^{2}\) | \(\left(-\dfrac{a^{2}m}{c},\dfrac{b^{2}}{c}\right)\) |
| \(\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1\) | \(c^{2}=a^{2}m^{2}-b^{2}\) | \(\left(-\dfrac{a^{2}m}{c},-\dfrac{b^{2}}{c}\right)\) |
H. Director circle (perpendicular pair of tangents)
- Parabola \(y^{2}=4ax\): the directrix \(x=-a\).
- Ellipse: \(x^{2}+y^{2}=a^{2}+b^{2}\).
- Hyperbola: \(x^{2}+y^{2}=a^{2}-b^{2}\).
I. Classifying \(Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\) (\(B=0\))
- \(A=C\): circle.
- Exactly one of \(A,C\) zero: parabola.
- \(A\neq C\), same sign: ellipse.
- \(A,C\) opposite signs: hyperbola.