conic-sections

Parabola

1. The fixed point S is called focus
2. The fixed straight line is called the directrix.
3. A line through the focus and perpendicular to the directrix is called the axis (or axis of symmetry) of the Parabola.
4. The point of intersection of the Parabola with its axis is called its vertex.
5. A chord perpendicular to the axis is called double ordinate.
6. The double ordinate through the focus is called the latus rectum.

• vertex at origin $y^2=4ax\text{right parabola}$ $y^2=-4ax\text{left parabola}$ $x^2=4ay\text{upper parabola}$ $x^2=-4ay\text{lower parabola}$

• vertex at any point $(h,k)$ $(y-k{)}^2=4a(x-h)\text{right parabola}$ $(y-k{)}^2=-4a(x-h)\text{left parabola}$ $(x-h{)}^2=4a(y-k)\text{upper parabola}$ $(x-h{)}^2=-4a(y-k)\text{lower parabola}$

• quadractic representation $x=a{y}^2+by+c\text{horizontal}$ $y=a{x}^2+bx+c\text{vertical}$

• following are true for all parabolas

  • distance between vertex and focus = $\frac{1}{4}\times \text{latus rectum}$
  • distance between directrix and latus rectum = $\frac{1}{2}\times \text{latus rectum}$
  • vertex is always the midpoint of focus and point of intersection of axis and directrix
  • distance of any point on parabola from it’s axis = $\sqrt{l\times t}$

here, l - latus rectum , t - distance of that point from tangent at vertex

Hyperbola

Property	Horizontal Hyperbola	Vertical Hyperbola
Equation	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
Equation with center at $(h,k)$	$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1,a>b$	$-\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1,b>a$
Parameteric form	$x=a\sec \theta y=b\tan \theta$	$x=a\sec \theta y=b\tan \theta$
Transverse Axis	Along X-axis, length = $2a$	Along Y-axis, length = $2b$
Conjugate Axis	Along Y-axis, length = $2b$	Along X-axis, length = $2a$
Eccentricity	$e=\sqrt{1+\frac{b^2}{a^2}}$	$e=\sqrt{1+\frac{a^2}{b^2}}$
Foci	$S(\pm ae,0)$	$S(0,\pm be)$
Directrices	$x=\pm \frac{a}{e}$	$y=\pm \frac{b}{e}$
Latus Rectum	$x=\pm ae$	$y=\pm be$
Length of Latus Rectum	$LR=\frac{2b^2}{a}$	$LR=\frac{2a^2}{b}$
Distance between focii	$d=2a\times e$	$d=2b\times e$
Distance between directrices	$d=\frac{2a}{e}$	$d=\frac{2b}{e}$

Ellipse

Property	Horizontal Ellipse	Vertical Ellipse
Equation	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,b>a$
Equation with center at $(h,k)$	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1,a>b$	$\frac{(x-h{)}^2}{a^2}+\frac{(y-k{)}^2}{b^2}=1,b>a$
Parameteric form	$x=a\cos \theta y=b\sin \theta$	$x=a\cos \theta y=b\sin \theta$
Major Axis	Along X-axis, length = $2a$	Along Y-axis, length = $2b$
Minor Axis	Along Y-axis, length = $2b$	Along X-axis, length = $2a$
Eccentricity	$e=\sqrt{1-\frac{b^2}{a^2}}$	$e=\sqrt{1-\frac{a^2}{b^2}}$
Foci	$S(\pm ae,0)$	$S(0,\pm be)$
Directrices	$x=\pm \frac{a}{e}$	$y=\pm \frac{b}{e}$
Latus Rectum	$x=\pm ae$	$y=\pm be$
Length of Latus Rectum	$LR=\frac{2b^2}{a}$	$LR=\frac{2a^2}{b}$
Distance between focii	$d=2a\times e$	$d=2b\times e$
Distance between directrices	$d=\frac{2a}{e}$	$d=\frac{2b}{e}$

Circles

• $x^2+y^2+2gx+2fy+c=0\text{center at (-g,-f) and radius = g2+f2−c}$

  • Intercept made by a circle on x-axis $|AB|=2\sqrt{g^2-c}$
  • Intercept made by a circle on y-axis $|AB|=2\sqrt{f^2-c}$

• $(x-x_1{)}^2+(y-y_1{)}^2=r^2\text{center at (x1,y1) }$

  • Parametric Equation of the circle $x=x_1+r\cos \theta y=y_1+r\cos \theta$

• diameteric equation of circle $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0\text{here, (x1,y1)−(x2,y2) is a diameter of the circle}$

Family Of Circles

S - equation of circle, L - equation of line

• Family of circles passing through the intersection of circle S=0 and line L=0 is represented as $S+\lambda L=0$
• Family of circles passing through the intersection of circle $S_1=0$ and another circle $S_2=0$ is represented as: (here, $\lambda \ne -1$) $S_1+\lambda S_2=0$

  here, if $\lambda =-1$ then $S_1-S_2=0$ will give us the equation of the common chord of both the circles

• Family of circles passing throught the points $(x_1,y_1)$ and $(x_2,y_2)$ is represented by $S+\lambda L=0$ $S=(x-x_1)(x-x_2)+(y-y_1)(y-y_2)$ $L=(y-y_1)-\frac{y_2-y_1}{x_2-x_1}(x-x_1)$
• Family of circles passing tangent to a given line $L=0$ at a given point $(x_1,y_1)$ is represented as $(x-x_1{)}^2+(y-y_1{)}^2+\lambda L=0$