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Co–Ordinate Geometry

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☼    POINT   ☼

► Distance Formula :–

\large \mapsto Distance between A and B is denoted by d(A, B) or AB. Distance formula is defined by as follow :

  • For R1 :– If  A(x) and B(y), then AB = \left|x-y\right|
  • For R2 :– If  A (x1, y1) and B ( x2, y2), then AB  =  \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}
  • For R3 :– If A (x1, y1, z1) and B ( x2, y2, z2), then

AB  =  \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2}

\large \mapsto Properties for Distance Formula :–

There are mainly four properties for distance formula, which are as follow :

  1. AB \geq 0
  2. AB =  0  \Leftrightarrow  A = B
  3. AB =  BA
  4. AB \le AC +  BC

► Shifting of  Origin ( Without changing the scale ) :–

Shifting the origin at (h, k), the new co–ordinates of  P(x, y) is P’ (x’, y’ ). And

x’  = x – h,  y’ = y – k

► Area of  a triangle :–

If  A (x1, y1), B ( x2, y2) and C ( x3, y3), then

Area of  \Delta ABC  =   ABC   =  \Delta  =  \frac{1}{2}\left|D\right| ;  Where,  D  =  \left|\begin{matrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\\\end{matrix}\right|

► Condition for collinearity :–

If  A (x1, y1), B ( x2, y2) and C ( x3, y3)  are  collinear  \Leftrightarrow   \left|\begin{matrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\\\end{matrix}\right|    =  0

► Division of  line – segment :–

\large \mapstoInternal Division :–

  • If P divides \overline{\mathrm{AB}} from A, internally, in the ratio \lambda , then \lambda =  \frac{\mathrm{AP}}{\mathrm{PB}}
  • A – P – B \Leftrightarrow \lambda > 0
  • Let, A (x1, y1) & B ( x2, y2). If P (x, y) divides \overline{\mathrm{AB}} from A in the ratio \lambda, then

x  =  \frac{\ {\lambda\ x}_2+x_1}{\lambda+1}   and   y =  \frac{\lambda\ \ y_2+y_1}{\lambda+1}

Let, A (x1, y1) & B ( x2, y2). If P (x, y) divides \overline{\mathrm{AB}}  from A in the ratio λ = m : n, then

x  =  \frac{\mathrm{m}\mathrm{x}_2+\mathrm{n} \mathrm{x}_1}{m+n}   and   y =  \frac{\mathrm{m}\mathrm{y}_2+\mathrm{n} \mathrm{y}_1}{m+n}

Internal Division :–

  • If P divides \overline{\mathrm{AB}} from A, internally, in the ratio λ, then  λ  =   – \frac{\mathrm{AP}}{\mathrm{PB}}
  • P – A – B ⇔ – 1 < λ < 0   &    A – B – P  ⇔ λ < – 1
  • Let, A (x1, y1) & B ( x2, y2). If P (x, y) divides \overline{\mathrm{AB}}  from A in the ratio λ, then

x  =  \frac{\lambda\ x_2+x_1}{\lambda+1}   and   y =  \frac{\lambda\ y_2+y_1}{\lambda+1}  ;  Where,  λ  ≠  – 1.

  • Let, A (x1, y1) &   B ( x2, y2).  If    P (x, y)  divides  \overline{\mathrm{AB}} from  A  in  the  ratio λ = m : – n, then

x  =  \frac{\mathrm{m}\mathrm{x}_2-\mathrm{n} \mathrm{x}_1}{m-n}   and   y =  \frac{\mathrm{m}\mathrm{y}_2-\mathrm{n} \mathrm{y}_1}{m-n}  ;  Where,  λ  ≠  – 1.

► Centres of  a triangle :–

↦ Circumcentre ( P ) :–

Let,  A (x1, y1),  B ( x2, y2)  and  C ( x3, y3).  If   P (x, y) is  a  circumcentre  of  Δ ABC, then   PA  =  PB  =  PC  =  R ( Circum–radius )

↦ Centroid ( G ) :–

Let,   A (x1, y1),  B ( x2, y2)  and  C ( x3, y3).  If   G (x, y)   is  a  in centroid  of  Δ ABC, then   x  =  \frac{x_1+x_2+x_2}{3}    &    y  =  \frac{y_1+y_2+y_2}{3}

↦ In centre ( I ) :–

Let,   A (x1, y1),  B ( x2, y2)  and  C ( x3, y3).  If   I (x, y)   is  a  in centre  of  Δ ABC, then   x  =  \frac{\mathrm{a}\mathrm{x}_1+\mathrm{b} \mathrm{x}_2+\mathrm{c} \mathrm{x}_2}{a+b+c}    &    y  =  \frac{\mathrm{a}\mathrm{y}_1+\mathrm{b} \mathrm{y}_2+\mathrm{c} \mathrm{y}_2}{a+b+c}

;   Where,   a  =  BC,   b  =  AC,    c  =  AB

↦ Orthocentre ( H ) :–

G divides  from P in the ratio 2 : 1.

 

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