Co-ordinate Geometry

1. Distance formula: the distance between the points ($x_1$, $y_1$) and ($x_2$, $y_2$) is given by $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

2. Fourth vertex of a parallelogram, If A($x_1$, $y_1$) B ($x_2$, $y_2$) C ($x_3$, $y_3$) are the 3 consecutive vertices of a triangle then the $4^{\mathrm{th}}$ vertex is ($x_1$ + $x_3$-$x_2$, $y_1$ + $y_3$-$y_2$)

3. Vertices, when mid points are given: If ($x_1$, $y_1$) ($x_2$, $y_2$) ($x_3$, $y_3$) are the mid points of the sides of a triangle then the vertices are ($x_1$ + $x_2$-$x_3$, $y_1$ + $y_2$-$y_3$) ; ($x_1$ + $x_3$-$x_2$, $y_1$ + $y_3$-$y_2$) and ($x_2$ + $x_3$-$x_1$, $y_2$ + $y_3$-$y_1$)

4. Section formula: The coordinates of a point P which divides the line joining the points

A($x_1$, $y_1$) and B ($x_2$, $y_2$) in the ratio l:m internally is $(\frac{{\mathrm{lx}}_2+{\mathrm{mx}}_1}{1+m},\frac{{\mathrm{ly}}_2+{\mathrm{my}}_1}{1+m})$

Note 1: If the division is externalthen P is $(\frac{{\mathrm{lx}}_2-{\mathrm{mx}}_1}{1-m},\frac{{\mathrm{ly}}_2-{\mathrm{my}}_1}{1-m})$

Note 2: Mid point of A($x_1$, $y_1$) and B ($x_2$, $y_2$) is $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

Note 3: Trisection means division in the ration 1:2 or in 2:1

Note 4: P(x,y) divides the join of ($x_1$, $y_1$) and ($x_2$, $y_2$) in the ratio l:m then the ratio

l:m= - (x - $x_1$) : (x - $x_2$)

Note 5: The ratio in which x axis divides the join of ($x_1$, $y_1$) and ($x_2$, $y_2$); l:m = -$\frac{y_1}{y_2}$

Note 6: The ratio in which y axis divides the join of ($x_1$, $y_1$) and ($x_2$, $y_2$); l:m = -$\frac{x_1}{x_2}$

6. Area of triangle: Area of a triangle with vertices ($x_1$, $y_1$), ($x_2$, $y_2$) and ($x_3$, $y_3$) is

1/2 {$x_1$($y_2$-$y_3$) + $x_2$($y_3$-$y_1$) + $x_3$($y_1$-$y_2$)} = $\frac{1}{2s}$ $\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

7. Area of a quadrilateral: Split it into 2 triangles and find the sum of their areas.

8. Centroid of triangle: where are ($x_1$, $y_1$); ($x_2$, $y_2$) and ($x_3$, $y_3$) is $[\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}]$

9. In- centre of a triangle: It is the point of intersection of internal bisectors of the angles of the triangle. If a,b,c are the length of sides the triangle opposite to the vertices A($x_1$, $y_1$) B($x_2$, $y_2$) and C ($x_3$, $y_3$) Then in-centre of triangle ABC is $(\frac{{\mathrm{ax}}_1+{\mathrm{bx}}_2+{\mathrm{cx}}_3}{a+b+c},\frac{{\mathrm{ay}}_1+{\mathrm{by}}_2+{\mathrm{cy}}_3}{a+b+c})$

10. Slope of a line:

If a line makes an angle $\theta$ with the +ve direction of the x axis then slope of the line= tan$\theta$

Note:-

1. slope of the x axis= 0

2. slope of a line parallel to x axis= 0

3. slope of the y axis = not defined

4. slope of a parallel to y axis= not defined

slope of the line segment joining the points ($x_1$, $y_1$) and ($x_2$, $y_2$)= $\frac{y_2-y_1}{x_2-x_1}$

11. Locus of a point:- It is the path traced by a point.

12. Eqn. of locus:- It is the relation satisfied by the x and y co-ordinates of any point on the locus.

13. Eqn. of a straight line:-

1. Eqn. of the x axis is y= 0

2. Eqn. of the y axis is x= 0

3. Eqn. of a horizontal line (line parallel to x axis) is y=k

4. Eqn. of a vertical line (line parallel to y axis) is x=k

5. Point slope form: If ($x_1$, $y_1$) is a point on th eline and if slope of the line is m then

eqn. of the line is: y - $y_1$= m (x - $x_1$)

6. Two Points from : Eqn of the line passing through the point ($x_1$, $y_1$) and ($x_2$, $y_2$) is

$\frac{y-y_1}{y_2-y_1}$= $\frac{x-x_1}{x_2-x_1}$

7. Eqn. of a line making intercepts a and b on the axis of Co-ordinates is $\frac{x}{a}$+$\frac{y}{b}$= 1

8. Slope y intercept form Eqn. of a line having a slope m and y intercept is given by y=

mx+c Perpendicular form or normal form:

If $\rho$ is the perpendicular distance of the line from the origin and if $\alpha$ is the angle, the

line makes with the positive direction of the x-axis, then its equation is x cos $\propto$ + y sin

$\propto$= $\rho$.

10. General form: The general form of the equation of a line is ax+by+c= 0 ie equation

of a straight line is a first degree equation in x and y. and conversely every first degree

equation in x and y represents a straight line.

14. Angles between two lines.'

The acute angle between two lines with slope $m_1$ and $m_2$ is given by tan$\theta$= $\left|\frac{m_2-m_1}{1+m_1{m}_2}\right|$

Note:-

1. If two lines are parallel then their slopes are equal and if the slopes are equal the

lines are parallel.

2. If the lines are perpendicular product of the slopes is -1. Conversely if $m_1{m}_2$= -1

then the lines are $\perp$r.

3. A line parallel to ax + by + c= 0 can be written as ax+by+k= 0

4. A line perpendicular to ax + by + c= 0 can be written as bx - ay + k= 0

5. If a line / curve passes through origin then constant term in its equation will be

absent.

15. Distance of a line form a point

The $\perp$ r distance between the point ($x_1$, $y_1$) and the line ax + by + c= 0 is given by

$\left|\frac{{\mathrm{ax}}_1+{\mathrm{by}}_1+c}{\sqrt{a^2+b^2}}\right|$

Note:- Distance of the origin from ax + by + c = 0 is given by $\left|\frac{c}{\sqrt{a^2+b^2}}\right|$

16. Distance between two parallel lines: Distance between the parallel lines ax + by +$K_1$= 0 and ax + by +$K_2$= 0 is given by $\left|\frac{K_1-K_2}{\sqrt{a^2+b^2}}\right|$


17. Line through the intersection of two lines: Equation a line through the intersection two lines $L_1$= 0 and $L_2$= 0 is $L_1$+ l $L_2$= 0 where k is a constant.

18. Circum Centre: It is the point of intersection of the perpendicular bisectors of the sides of a triangle. To find circum centre, find any two perpendicular bisectors and solve them. OR: If O is the circum centre of the triangle ABC then OA = OB = OC and verify this codition.

Note:- If $\mathrm{\Delta }$ ABC is right angled then circum centre = mid point of the hypotenuse:

19. Ortho centre: It is the point of intersection of altitudes of the triangles. To find it, find the equation of any two altitudes and solve them.

Note:-

1. for a right angled triangle ortho centre = The vertex opposite to the hypotenuse.

2. The circum centre C, the centroid G and orthocentre O of a triangle ABC are collinear and G divides OC in the ration 2:1 ie. $\frac{\mathrm{OG}}{\mathrm{GC}}$= 2:1

Pair of lines

1. Joint equation of pair of straight lines:

If $a_1$x + $b_1$y + $c_1$= 0 and $a_2$x + $b_2$y + $c_2$= 0 are two straight lines then their joint

equation is given by ($a_1$x + $b_1$y+ $c_1$) ($a_2$x + $b_2$y+ $c_2$)= 0. This equation represents 2

lines together or a pair of lines.


2. Pair of straight lins through origin :

A homogeneous equation of second degree ${\mathrm{ax}}^2$ + ${\mathrm{by}}^2$ + 2 hxy= 0 represents a pair of

lines through origin if $h^2$ $\ge$ ab:

3. Angle between ${\mathrm{ax}}^2$ + ${\mathrm{by}}^2$ + 2hxy = 0: is given by tan$\theta$= $\frac{2\sqrt{h^2-\mathrm{ab}}}{a+b}$

Note:

1. The lines are perpendicular if a+b= 0

2. If $h^2$ - ab= 0 the lines are coincident

3. If $h^2$ $<$ ab the lines are imaginary.

4. Sum & product of slopes:

If y= $m_1$x and y= $m_2$x are the separate equation of the lines ${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy= 0; then

$m_1$ + $m_2$= $\frac{-2h}{b}$ and $m_1$ $m_2$= a/b

5. Equation of bisectors:

The joint equation of the pair of bisectors of the angles between the lines given by

${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy= 0 is given by $\frac{x^2-y^2}{a-b}$= $\frac{\mathrm{xy}}{h}$

6. Equation of perpendicular pair:

The equation of 2 lines through origin perpendicular to ${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy= 0 is given by ${\mathrm{bx}}^2$+ ${\mathrm{ay}}^2$- 2hxy= 0

7. The general second degree equation: ${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy + 2gx + 2fy + c= 0 represents a pair of lines if

$\left|\begin{array}{ccc}a& h& g\\ h& b& f\\ g& f& c\end{array}\right|$ = 0 ie abc+ 2fgh - ${\mathrm{af}}^2$- ${\mathrm{bg}}^2$ - ${\mathrm{ch}}^2$= 0 and $h^2$ $\ge$ ab

8. If ${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy + 2gx + 2fy + c= 0 represents a pair of lines then:

1. Equation of lines through origin and parallel to this, is ${\mathrm{ax}}^2$+ ${\mathrm{by}}^2$+2hxy= 0.

2. Angle between them is given by tan$\theta$= $\frac{2\sqrt{h^2}}{a+b}$

3. Their point of intersection is $(\frac{\mathrm{hf}-\mathrm{bg}}{\mathrm{ab}-h^2},\frac{\mathrm{hg}-\mathrm{af}}{\mathrm{ab}-h^2})$

4. The equation of bisectors of the angle between the is given by $\frac{{(x-x_1)}^2-{(y-y_1)}^2}{a-b}$=

$\frac{(x-x_1)(y-y_1)}{h}$ where ($x_1,y_1$) is the point of intersection.

5. They are parallel if a/h = h/b = g/f and distance between these parallel lines = 2

$\sqrt{\left(\frac{g^2-\mathrm{ac}}{a(a+b)}\right)}$ .