Coordinate Geometry – Quick Revision Notes

Cartesian System of Rectangular Coordinates

• Two mutually perpendicular axes $XOX'$ (x–axis) & $YOY'$ (y–axis) meet at the origin $O(0,0)$.

• Coordinates of a point $P(x,y)$:
  • Abscissa = $x$ (distance from $y$–axis)
  • Ordinate = $y$ (distance from $x$–axis)

• Quadrant sign convention:
  • I (x>0,y>0) • II (x<0,y>0) • III (x<0,y<0) • IV (x>0,y<0)

Distance & Its Applications

• Distance between $P(x<em>1,y</em>1)$ & $Q(x<em>2,y</em>2)$
  $PQ = \sqrt{(x<em>2-x</em>1)^2+(y<em>2-y</em>1)^2}$

• Distance from origin: $OP = \sqrt{x^2+y^2}$.

• Pythagorean tests:
  • Right–triangle: $PQ^2+QR^2=PR^2$
  • Collinear points: Sum of two smaller distances equals the third.

Section Formulae

• Internal division of $PQ$ in ratio $m<em>1:m</em>2$ by $R(x,y)$:
  $x = \dfrac{m<em>1x</em>2+m<em>2x</em>1}{m<em>1+m</em>2},\; y = \dfrac{m<em>1y</em>2+m<em>2y</em>1}{m<em>1+m</em>2}$

• Mid-point $(m<em>1=m</em>2)$: $\bigl(\dfrac{x<em>1+x</em>2}{2},\dfrac{y<em>1+y</em>2}{2}\bigr)$

• External division (same ratio):
  $x = \dfrac{m<em>1x</em>2-m<em>2x</em>1}{m<em>1-m</em>2},\; y = \dfrac{m<em>1y</em>2-m<em>2y</em>1}{m<em>1-m</em>2}$

Area & Collinearity

• Area of $\triangle ABC$ with vertices $A(x<em>1,y</em>1),B(x<em>2,y</em>2),C(x<em>3,y</em>3)$:
  $\text{Area}=\dfrac12\left|x<em>1(y</em>2-y<em>3)+x</em>2(y<em>3-y</em>1)+x<em>3(y</em>1-y_2)\right|$

• Points collinear $\Leftrightarrow$ area $=0$ $\Rightarrow$ determinant condition
  \begin{vmatrix}x1&y1&1\x2&y2&1\x3&y3&1\end{vmatrix}=0

Inclination & Slope of a Line

• Inclination $\theta$: angle w.r.t. positive x-axis (anticlockwise).

• Slope $m=\tan\theta$ (undefined for vertical lines).

• Slope of line through $P<em>1(x</em>1,y<em>1),P</em>2(x<em>2,y</em>2)$:
  $m=\dfrac{y<em>2-y</em>1}{x<em>2-x</em>1}$ (vertical $\Rightarrow x<em>1=x</em>2$).

• Special results:
  • Parallel lines $m<em>1=m</em>2$
  • Perpendicular lines $m<em>1m</em>2=-1$
  • Angle between two lines $\tan\phi = \dfrac{|m<em>1-m</em>2|}{1+m<em>1m</em>2}$

Straight–Line Equations

1. Slope–Intercept

$y=mx+c$ (slope $m$, y-intercept $c$).

2. Point–Slope

Through $P(x<em>1,y</em>1)$: $y-y<em>1=m(x-x</em>1)$.

3. Two-Point

Through $P<em>1,P</em>2$: $\dfrac{y-y<em>1}{y</em>2-y<em>1}=\dfrac{x-x</em>1}{x<em>2-x</em>1}$.

4. Intercept Form

$\frac{x}{a}+\frac{y}{b}=1$ (cuts axes at $a, b$).

5. Normal (Perpendicular) Form

$x\cos\alpha + y\sin\alpha = p$
$p$ = perpendicular from origin, $\alpha$ its inclination.

6. General First-Degree

$Ax+By+C=0$ represents a line if $A,B$ not both zero.
• Slope $= -\dfrac AB$, x-intercept $= -\dfrac CA$, y-intercept $= -\dfrac CB$.

Distance of Point from Line

For point $P(x<em>1,y</em>1)$ & line $Ax+By+C=0$:
$d = \dfrac{|Ax<em>1+By</em>1+C|}{\sqrt{A^2+B^2}}$

Shift of Origin (Translation of Axes)

• New origin $O'(h,k)$: $x=x'+h,\; y=y'+k$.

• Any curve $F(x,y)=0$ transforms to $F(x'+h,y'+k)=0$.

Families of Lines

• Lines through intersection of $l<em>1=0$ & $l</em>2=0$: $l<em>1+\lambda l</em>2=0$.

• Parallel to given line $Ax+By+C=0$: $Ax+By+K=0$.

• Perpendicular to it: $Bx-Ay+K=0$.

Circles

• Standard form (centre $h,k$ radius $r$): $(x-h)^2+(y-k)^2 = r^2$.

• General second-degree with equal $x^2,y^2$ coefficients & no $xy$ term:
  $x^2+y^2+2gx+2fy+c=0$
  • Centre $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$.

• Special positions:
  • Touches x-axis $\Rightarrow k=r$ • Touches y-axis $\Rightarrow h=r$ • Touches both $\Rightarrow h=k=r$.

Conic Sections (Eccentricity Definition)

Given focus $S$, directrix $l$, eccentricity $e$.

• Circle $e=0$ (special ellipse).

• Ellipse 0<e<1.

• Parabola $e=1$.

• Hyperbola e>1.

Ellipse (Standard)

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\;(a>b>0)

• $c^2=a^2-b^2,\; e=\dfrac ca$.

• Foci $(\pm c,0)=(\pm ae,0)$.

• Directrices $x=\pm\dfrac a e$.

• Major axis $2a$, minor axis $2b$.

• Length of latus rectum $=\dfrac{2b^2}{a}$.
  (Interchange $a,b$ for vertical major axis.)

Parabola (Standard)

$y^2=4ax$

• Vertex $(0,0)$, focus $(a,0)$, directrix $x=-a$.

• Axis along x-axis; latus rectum $4a$.
  Other orientations: replace $y^2$ by $x^2$ and/or change signs.

Hyperbola (Standard)

\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\;(a,b>0)

• c^2=a^2+b^2,\; e=\dfrac ca>1.

• Foci $(\pm c,0)=(\pm ae,0)$.

• Directrices $x=\pm\dfrac a e$.

• Transverse axis $2a$, conjugate axis $2b$.

• Latus rectum $=\dfrac{2b^2}{a}$.
  Conjugate hyperbola: $\dfrac{y^2}{b^2}-\dfrac{x^2}{a^2}=1$.
  Rectangular hyperbola: $a=b\Rightarrow e=\sqrt2$.

Key Problem-Solving Steps

1. Identify type via $e$ or general equation criteria.

2. Normalize coefficients to match standard forms.

3. Extract parameters $a,b,c,e$, then derive foci, axes lengths, directrices, latus rectum.

4. For locus problems use definition $\dfrac{\text{(distance to focus)}}{\text{(distance to directrix)}}=e$.

End of Module IV Revision Notes – Coordinate Geometry (Straight Lines & Conics)