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Formulas: analytic geometry, grade 12 (5 units)

Points and lines

  • The distance between the points (x1,y1)\displaystyle (x_1,y_1) and (x2,y2)\displaystyle (x_2,y_2)
    d\displaystyle d =\displaystyle = (x1−x2)2+(y1−y2)2\displaystyle \sqrt{(x_1-x_2)^{2}+(y_1-y_2)^{2}}
  • The slope of the line through the two points
    m\displaystyle m =\displaystyle = y2−y1x2−x1\displaystyle \frac{y_2-y_1}{x_2-x_1}x1≠x2\displaystyle x_1\neq x_2
  • The line with slope m\displaystyle m through (x1,y1)\displaystyle (x_1,y_1)
    y−y1=m(x−x1)\displaystyle y-y_1=m(x-x_1)
  • Perpendicular lines – m1\displaystyle m_1, m2\displaystyle m_2 their slopes
    m1⋅m2\displaystyle m_1\cdot m_2 =\displaystyle = −1\displaystyle -1
  • The point C\displaystyle C dividing the segment AB\displaystyle AB in the ratio AC:CB=k:l\displaystyle {AC:CB}={k:l} – A(x1,y1)\displaystyle A(x_1,y_1), B(x2,y2)\displaystyle B(x_2,y_2)
    C(lx1+kx2k+l,ly1+ky2k+l)\displaystyle C\left(\frac{lx_1+kx_2}{k+l},\frac{ly_1+ky_2}{k+l}\right)
  • The distance of the point (x0,y0)\displaystyle (x_0,y_0) from the line Ax+By+C=0\displaystyle {Ax+By+C}={0}
    d\displaystyle d =\displaystyle = ∣Ax0+By0+C∣A2+B2\displaystyle \frac{|Ax_0+By_0+C|}{\sqrt{A^{2}+B^{2}}}
  • The distance between the parallel lines Ax+By+C1=0\displaystyle {Ax+By+C_1}={0} and Ax+By+C2=0\displaystyle {Ax+By+C_2}={0}
    d\displaystyle d =\displaystyle = ∣C1−C2∣A2+B2\displaystyle \frac{|C_1-C_2|}{\sqrt{A^{2}+B^{2}}}

Circle

  • A circle with centre (a,b)\displaystyle (a,b) and radius R\displaystyle R
    (x−a)2+(y−b)2=R2\displaystyle (x-a)^{2}+(y-b)^{2}=R^{2}
  • The expanded equation – a circle when, after completing the square, the right side is positive
    x2+y2+Cx+Dy+E=0\displaystyle x^{2}+y^{2}+Cx+Dy+E=0
  • The tangent to the circle at the point (x0,y0)\displaystyle (x_0,y_0) on it
    (x0−a)(x−a)+(y0−b)(y−b)=R2\displaystyle (x_0-a)(x-a)+(y_0-b)(y-b)=R^{2}

Parabola

  • The canonical parabola
    y2=2px\displaystyle y^{2}=2px
  • The focus
    F(p2,0)\displaystyle F\left(\frac{p}{2},0\right)
  • The directrix
    x=−p2\displaystyle x=-\frac{p}{2}
  • The tangent at the point (x0,y0)\displaystyle (x_0,y_0) on the parabola
    y⋅y0=p(x+x0)\displaystyle y\cdot y_0=p(x+x_0)

Ellipse

  • The canonical ellipse – the foci on the x\displaystyle x-axis when a>b\displaystyle a>b
    x2a2+y2b2=1\displaystyle \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  • The distance of a focus from the origin
    c\displaystyle c =\displaystyle = a2−b2\displaystyle \sqrt{a^{2}-b^{2}}
  • The sum of the distances of a point P\displaystyle P on the ellipse from the foci
    PF1+PF2\displaystyle PF_1+PF_2 =\displaystyle = 2a\displaystyle 2a

Hyperbola

  • The canonical hyperbola – the foci on the x\displaystyle x-axis
    x2a2−y2b2=1\displaystyle \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  • The distance of a focus from the origin
    c\displaystyle c =\displaystyle = a2+b2\displaystyle \sqrt{a^{2}+b^{2}}
  • The difference of the distances of a point P\displaystyle P on the hyperbola from the foci
    ∣PF1−PF2∣\displaystyle |PF_1-PF_2| =\displaystyle = 2a\displaystyle 2a
  • The asymptotes
    y=±bax\displaystyle y=\pm\frac{b}{a}x