Straight-line Formula Sheet

A straight line in geometry represents the shortest distance between two points and is defined by its constant slope and linear equation, y=mx+b. It extends infinitely in both directions without any curvature.

Neetesh Kumar | May 06, 2024 $\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space \space$ Share this Page on:

• 1. Relation Between Cartesian & Polar Coordinate System
• 2. Distance Formula and its Applications
• 3. Section Formula
• 4. Coordinates of Most commonly used Points
• 5. Area of Triangle
• 6. Condition of Collinearity of three Points
• 7. Equation of Straight Line
• 8. Slope of Line
• 9. Standard forms of equations of Straight Line
• 10. Angle between two Lines
• 11. Length of Perpendicular from a Point on a Line
• 12. Distance between two Parallel lines
• 13. Equation of Lines Parallel & Perpendicular to a given Line
• 14. Straight line making an angle with a given Line
• 15. Position of Two points concerning a given Line
• 16. Concurrency of Lines
• 17. Reflection of a Point
• 18. Transformation of Axes
• 19. Equation of Bisectors of Angles between two lines
• 20. Family of Lines
• 21. General equation and Homogeneous equation of second degree
• 22. Equation of Lines joining the points of Intersection of a Line and a Curve to the Origin
• 23. Standard Results

1. Relation Between Cartesian & Polar Coordinate system:

If (x,y) are Cartesian coordinates of a point P, then x =r cos$\theta$, y = r sin$\theta$ and r = $\sqrt{x^2 + y^2}$, $\theta$ = $tan^{-1}(\frac{y}{x})$

2. Distance Formula and its Applications:

If A(x$_1$, y$_1$) and B(x$_2$, y$_2$) are two points, then AB = $\sqrt{(x_2 -x_1)^2 + (y_2 - y_1)^2}$
$\bold{Note:}$
$\bold{(i)}$ Three given points A, B, and C are collinear when the sum of any two distances out of AB, BC, and CA is equal to the remaining third; otherwise, the points will be the triangle's vertices.
$\bold{(ii)}$ Let A,B,C & D be the four given points in a plane. Then, the quadrilateral will be :

• Square if AB = BC = CD = DA & AC = BD ; AC $\perp$ BD
• Rhombus if AB = BC = CD = DA and AC $\ne$ BD; AC $\perp$ BD
• Parallelogram if AB = DC, BC = AD; AC $\ne$ BD; AC $\nvdash$ BD
• Parallelogram if AB = DC, BC = AD; AC = BD; AC $\nvdash$ BD

3. Section Formula :

The coordinates of a point dividing a line joining the points A (x$_1$, y$_1$) and B (x$_2$, y$_2$) in the ratio m:n is given by : $\bold{(a) \space Internal \space division:}$ $\bigg( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\bigg)$

$\bold{(b) \space External \space division:}$ $\bigg( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n}\bigg)$

$\bold{(c)}$ Line ax + by + c = 0 divides line joining points P(x$_1$, y$_1$) & Q(x$_2$, y$_2$) in ratio = $-\frac{(ax_1 + by_1 + c)}{(ax_2 + by_2 + c)}$

4. Coordinates of Most commonly used Points:

Let A (x$_1$, y$_1$) and B (x$_2$, y$_2$) are vertices of any triangle ABC, then
$\bold{(a) \space Centroid:}$

$\bold{(iv)}$ Co-ordinates of centroid G $\bigg( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\bigg)$

$\bold{(b) \space Incenter:}$

Co-ordinates of incenter I = $\bigg( \frac{ax_1 + bx_2 + cx_3}{a + b + c}, \frac{ay_1 + by_2 + cy_3}{a + b + c}\bigg)$

$\bold{Note:}$
$\bold{(i)}$ Angle bisector divides the opposite sides in the ratio of the remaining sides. e.g. $\frac{BD}{DC}=\frac{AB}{AC}=\frac{c}{b}$
$\bold{(ii)}$ Incenter divides the angle bisectors in the ratio (b + c) : a, (c + a) : b, (a + b) : c

$\bold{(c) \space Circumcenter:}$

$\bold{Note:}$
$\bold{(i)}$ If a triangle is a right angle, then its circumcenter is the midpoint of the hypotenuse.
$\bold{(ii)}$ Find the perpendicular bisector of any two sides and solve them to find the circumcentre.

$\bold{(d) \space Orthocenter:}$

$\bold{Note:}$
$\bold{(i)}$ If a triangle is a right-angled triangle, then the orthocenter is where the right angle is formed.
$\bold{(ii)}$ If the triangle is equilateral, then centroid, incentre, orthocenter, and circumcenter coincide.

$\bold{(iii)}$ Orthocentre, centroid, and circumcentre are always collinear, and the centroid divides the line joining orthocentre and circumcentre in the ratio 2:1
$\bold{(iv)}$ In an isosceles triangle, the centroid, orthocentre, incentre, and circumcentre lies on the same line.

$\bold{(e) \space Excenter:}$

$I_1 = \bigg( \frac{-ax_1 + bx_2 + cx_3}{-a + b + c}, \frac{-ay_1 + by_2 + cy_3}{-a + b + c}\bigg)$

Similarly, ex-centers of $\Delta$ ABC with respect to vertices B and C are denoted by $I_2$ and $I_3$, respectively, and

$I_2 = \bigg( \frac{ax_1 - bx_2 + cx_3}{a - b + c}, \frac{ay_1 - by_2 + cy_3}{a - b + c}\bigg),$

$I_3 = \bigg( \frac{ax_1 + bx_2 = cx_3}{a + b - c}, \frac{ay_1 + by_2 - cy_3}{a + b - c}\bigg)$

5. Area of Triangle:

Let A(x$_1$, y$_1$), B(x$_2$, y$_2$) and C(x$_3$, y$_3$) are vertices of a triangle, then Area of $\Delta$ ABC = |$\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}$| = $\frac{1}{2}|[x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)]|$
To remember the above formula, take the help of the following method : = $\frac{1}{2}|[(x_1y_2 - x_2y_1) + (x_2y_3 - x_3y_2) + (x_3y_1 - x_1y_3)]|$
$\bold{Note:}$
$\bold{(i)}$ If the area of the triangle joining three points is zero, then the points are collinear.
$\bold{(ii)}$ If the altitude of any equilateral triangle is P, then its area = $\frac{P^2}{\sqrt{3}}$
If 'a' be the side of equilateral triangle, then its area = $\frac{a^2\sqrt{3}}{4}$
$\bold{(iii)}$ Area of quadrilateral whose consecutive vertices are (x$_1$, y$_1$), (x$_2$, y$_2$), (x$_3$, y$_3$), (x$_4$, y$_4$) is $\frac{1}{2}\begin{vmatrix} x_1 - x_3 & y_1 - y_3 \\ x_2 - x_4 & y_2 - y_4 \\ \end{vmatrix}$

6. Condition of Collinearity of three Points:

Three points (x$_1$, y$_1$), (x$_2$, y$_2$), and (x$_3$, y$_3$) are collinear if any one of the the given points lie on the line passing through the remaining two points.
Thus, the required condition is:
$\frac{y_3 - y_1}{y_2 - y_1}$ = $\frac{x_3 - x_1}{x_2 - x_1}$ or $\frac{x_1 - x_2}{x_1 - x_3}$ = $\frac{y_1 - y_2}{y_1 - y_3}$ or $\frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}$ = 0

7. Equation of Straight Line:

A relation between x and y, which is satisfied by the coordinates of every point on a line, is called an equation of the straight line. Remember that every one-degree equation in variables x and y always represents a straight line, i.e., ax + by + c = 0; a & b $\ne$ 0 simultaneously.
$\bold{(a)}$ Equation of a line parallel to the x-axis at a distance a is y = a or y = – a
$\bold{(b)}$ Equation of x-axis is y = 0
$\bold{(c)}$ Equation of line parallel to y-axis at a distance b is x = b or x = – b
$\bold{(d)}$ Equation of y-axis is x = 0

8. Slope of Line:

If a given line makes an angle $\theta$ (0° $\le$ $\theta$ $\le$ 180°, $\theta$ $\ne$ 90°)

If A (x$_1$, y$_1$) and B (x$_2$, y$_2$) & $x_1 \ne x_2$ then slope of line AB = $\frac{y_2 - y_1}{x_2 - x_1}$

9. Standard forms of Equation of a Straight Line :

(a) Slope Intercept form:

Let m be the slope of a line and c its intercept on the y-axis. Then, the equation of this straight line is written as $\bold{y = mx + c}$

(b) Point Slope form:

If m is the slope of a line and it passes through a point (x$_1$, y$_1$), then its equation is written as $\bold{y – y_1 = m(x – x_1)}$

(c) Two-Point form:

Equation of a line passing through two points (x$_1$, y$_1$) and (x$_2$, y$_2$) is written as : $\bold{y – y_1 = (\frac{y_2 - y_1}{x_2 - x_1})(x – x_1)}$ or $\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \\ \end{vmatrix}$ = 0

(d) Intercept form:

If a and b are the intercepts made by a line on the axes of x and y, its equation is written as : $\bold{\frac{x}{a} + \frac{y}{b} =1}$

(e) Normal form:

If p is the length of the perpendicular on a line from the origin and an angle that this perpendicular makes with a positive x-axis, then the equation of this line is written as : $\bold{xcos\alpha + ysin\alpha = p}$ (p is always positive), where 0 $\le$ $\alpha$ < 2$\pi$

(f) Parametric form:

Let AP = r then x – h = r cos$\theta$, y – k = rsin$\theta$ & $\frac{x-h}{cos\theta} = \frac{y-k}{sin\theta}$ = r is the equation of the straight line LAL'.
Any point P on the line will be of the form (h + r cos$\theta$, k + r sin$\theta$), where | r | gives the distance of the point P from the fixed point (h, k).

(g) General form:

We know that a first-degree equation in x and y, $\bold{ax + by + c = 0}$, always represents a straight line. This form is known as a general form of straight line.
$\bold{(i)}$ Slope of this line = -$\frac{a}{b}$ = -$\frac{coeff. \space of \space x}{coeff. \space of \space y}$
$\bold{(ii)}$ Intercept made by this line on x-axis = -$\frac{c}{a}$ and intercept made by this line on y-axis = -$\frac{c}{b}$
$\bold{(iii)}$ To change the general form of a line to normal form, first take c to the right-hand side and make it positive, then divide the whole equation by $\sqrt{(a^2 + b^2)}$

10. Angle between two Lines:

$\bold{(a)}$ If $\theta$ be the angle between two lines : y = m$_1$x + c$_1$ and y = m$_2$x + c$_2$ then tan$\theta$ = $\plusmn(\frac{m_1 - m_2}{1 + m_1m_2})$
$\bold{(b)}$ If equation of lines are $a_1x+b_1y+c_1 =0$ and $a_2x+b_2y+c_2 =0$ then these lines are:

• Parallel $\iff$ $\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$
• Perpendicular $\iff$ $a_1a_2+b_1b_2=0$
• Coincident $\iff$ $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Intersecting $\iff$ $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$

11. Length of Perpendicular from a Point on a Line:

Length of perpendicular from a point (x$_1$, y$_1$) on the line ax + by + c = 0 is = |$\frac{ax_1+by_1+c}{\sqrt{a^2+b^2}}$|.
In particular, the length of the perpendicular from the origin on the line ax + by + c = 0 is P = $\frac{|c|}{\sqrt{a^2+b^2}}$.

12. Distance between two Parallel lines:

$\bold{(a)}$ The distance between two parallel lines ax + by + c$_1$ = 0 and ax + by + c$_2$ = 0 is $\frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}}$
$\bold{Note:}$ The coefficients of x & y in both equations should be same.
$\bold{(b)}$ The area of the parallelogram = $\frac{p_1p_2}{sin\theta}$, where $p_1$ & $p_2$ are distances between two pairs of opposite sides & q is the angle between any two adjacent sides. Note that the area of the parallelogram is bounded by the lines y = m$_1$x + c$_1$, y = m$_1$x + c$_2$ and y = m$_2$x + d$_1$, y = m$_2$x + d$_2$ is given by |$\frac{(c_1-c_2)(d_1-d_2)}{m_1-m_2}$|.

13. Equation of Parallel & Perpendicular lines to a given Line:

$\bold{(a)}$ Equation of line parallel to line ax + by + c = 0 is ax + by + $\lambda$ = 0
$\bold{(b)}$ Equation of line perpendicular to line ax + by + c = 0 is bx – ay + k = 0.
Here $\lambda$, k, are parameters, and their values are obtained with the Additional information was given on the problem.

14. Straight line making a given angle with a Line:

Equations of lines passing through a point (x$_1$, y$_1$) and making an angle a, with the line y=mx+c, is written as : $\bold{y-y_1=\frac{m \plusmn tan\alpha}{1∓ m.tan\alpha}}(x-x_1)$

15. Position of Two points with respect to a given Line:

Let the given line be ax + by + c = 0 and P(x$_1$, y$_1$), Q(x$_2$, y$_2$) be two points.
If the quantities ax$_1$ + by$_1$ + c and ax$_2$ + by$_2$ + c have the same signs, then both the points P and Q lie on the same side of the line ax + by + c = 0.
If the quantities ax$_1$ + by$_1$ + c and ax$_2$ + by$_2$ + c have opposite signs, they lie on the opposite sides of the line.

16. Concurrency of Lines:

Three lines a$_1$x + b$_1$y + c$_1$ = 0; a$_2$x + b$_2$y + c$_2$ = 0 and a$_3$x + b$_3$y + c$_3$ = 0 are concurrent if $\Delta$ = $\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{vmatrix}$ = 0 $\bold{Note:}$ If lines are concurrent then $\Delta$ = 0 but if $\Delta$ = 0 then lines may or may not be concurrent {lines may be parallel}.

17. Reflection of a Point:

18. Transformation of Axes:

(a) Shifting of origin without rotation of axes:

(b) Rotation of axes without shifting the origin:

The above relation between (x, y) and (x', y') can be easily obtained with the help of the following table

New/Old	X $\downarrow$	Y $\downarrow$
X' $\rightarrow$	cos $\theta$	sin $\theta$
Y' $\rightarrow$	-sin $\theta$	cos $\theta$

19. Equation of Bisectors of Angles between two lines:

If the equation of two intersecting lines is a$_1$x + b$_1$y + c$_1$ = 0 and a$_2$x + b$_2$y + c$_2$ = 0, then the equation of bisectors of the angles between these lines are written as:
$\frac{a_1x+b_1y+c_1}{\sqrt{{a_1}^2 + {b_1}^2}} = \plusmn\frac{a_2x+b_2y+c_2}{\sqrt{{a_2}^2 + {b_2}^2}}$ .....(1)

(a) Equation of Bisector of angle containing origin:

If the equation of the lines is written with constant terms $c_1$ and $c_2$ positive, then the equation of the bisectors of the angle containing the origin is obtained by taking the positive sign in (1)

(b) Equation of bisector of acute/obtuse angles:

See whether the two equations' constant terms c1 and c2 are +ve. If not, multiply both sides of the given equation by –1 to make the constant terms positive.
Determine the sign of $a_1a_2 + b_1b_2$

Sign of $a_1a_2 + b_1b_2$	For obtuse angle bisector	For acute angle bisector
+	Use + sign in equation (1)	Use - sign in equation (1)
-	Use - sign in equation (1)	Use + sign in equation (1)

i.e. if $a_1a_2 + b_1b_2$ > 0, then the bisector corresponding to + sign gives obtuse angle bisector $\frac{a_1x+b_1y+c_1}{\sqrt{{a_1}^2 + {b_1}^2}} = \frac{a_2x+b_2y+c_2}{\sqrt{{a_2}^2 + {b_2}^2}}$

20. Family of Lines:

If the equation of two lines is P = $a_1x + b_1y + c_1 = 0$ and Q = $a_2x + b_2y + c_2 = 0$, then the equation of the lines passing through the point of intersection of these lines is P + $\lambda$Q = 0 or $(a_1x + b_1y + c_1) + \lambda$ $(a_1x + b_1y + c_1)$ = 0. The value of $\lambda$ is obtained with the help of the additional information given in the problem.

21. General equation and Homogeneous equation of second degree:

$\bold{(a)}$ A general equation of second degree $\bold{ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0}$ represent a pair of straight lines if $\bold{\Delta =abc + 2fgh - af^2 -bg^2 - ch^2 = 0}$ or $\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \\ \end{vmatrix}$ = 0

$\bold{(b)}$ If $\theta$ be the angle between the lines, then tan$\theta$ = $\plusmn\frac{2\sqrt{h^2 - ab}}{a+b}$
$\space \space \space \space \space \space$ If these lines are:
$\space \space \space \space \space \space$ $\bold{(i)}$ Parallel, if $\Delta=0, h^2 = ab$ or if $h^2 = ab$ and $bg^2 = af^2$
$\space \space \space \space \space \space$ $\bold{(ii)}$ Perpendicular, if a + b = 0 i.e. coeff. of $x^2$ + coeff. of $y^2$ = 0.

$\bold{(c)}$ Homogeneous equation of $2^{nd}$ degree $ax^2 + 2hxy + by^2 = 0$ always represent a pair of straight lines whose equations are y = $(\frac{-h\plusmn\sqrt{h^2-ab}}{b})$x $\equiv$ y = m$_1$x & y = m$_2$x
$m_1 + m_2 = -\frac{2h}{b}$; $m_1m_2 = \frac{a}{b}$ These straight lines passes through the origin, and for finding the angle between these lines, the same formula as given for the general equation is used.
The conditions that these lines are :
$\space \space \space \space \space \space$ $\bold{(i)}$ At right angles to each other is a + b = 0. i.e. co-efficient of x² + co-efficient of y² = 0.
$\space \space \space \space \space \space$ $\bold{(ii)}$ Coincident is h² = ab.
$\space \space \space \space \space \space$ $\bold{(iii)}$ Equally inclined to the axis of x is h = 0 . i.e. coeff. of xy = 0.

$\bold{(d)}$ The combined equation of angle bisectors between the lines represented by a homogeneous equation of 2$^{nd}$ degree is given by $\frac{x^2 - y^2}{a-b}=\frac{xy}{h}, a \ne b, h \ne 0$

$\bold{(e)}$ Pair of straight lines perpendicular to the lines $ax^2 + 2hxy + by^2 = 0$ and through origin are given by $bx^2 – 2hxy + ay^2 = 0$.

$\bold{(f)}$ If lines $ax^2+ 2hxy + by^2 + 2gx + 2fy + c = 0$ are parallel then distance between them is $2\sqrt{\frac{g^2-ac}{a(a+b)}}$

22. Equation of Lines joining the points of Intersection of a Line and a Curve to the Origin:

$ax^2+ 2hxy + by^2 + 2(gx + fy)(\frac{lx+my}{-n}) + c(\frac{lx+my}{-n})^2 = 0$

23. Standard Results:

$\bold{(a)}$ Area of rhombus formed by lines a| x | + b| y | + c = 0 or $\plusmn$ ax $\plusmn$ by + c = 0 is $\frac{2c^2}{|ab|}.$

$\bold{(b)}$ Area of triangle formed by line ax + by + c = 0 and axes is $\frac{c^2}{2|ab|}.$

$\bold{(c)}$ Co-ordinate of the foot of perpendicular (h, k) from (x$_1$, y$_1$) to the line ax + by + c = 0 is given by
$\frac{h-x_1}{a} = \frac{k-y_1}{b} = \frac{-(ax_1+by_1+c)}{a^2+b^2}.$

$\bold{(d)}$ Image of point (x$_1$, y$_1$) w.r. to the line ax + by + c = 0 is given by
$\frac{h-x_1}{a} = \frac{k-y_1}{b} = \frac{-2(ax_1+by_1+c)}{a^2+b^2}.$

$\color{red} \bold{Related \space Pages:}$
Coordinate Geometry Calculators