Question

straight lines lesson all formulas fast guys plz army I need it fast ​

Answer 1

I'll cover all the important formulas and important concepts in this answer according to CBSE class-11 syllabus.

Distance formula

We use distance formula to find the distance between two given points.

In 2-D geometry

Let's say we have two given points (x1, y1) and (x2, y2) their distance will be given by,

• $\sf Distance = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

In 3-D geometry

Let's say we have two given points (x1, y1, z1) and (x2, y2, z2), their distance will be given by,

• $\sf Distance = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2 + (z_2-z_1)^2}$

Centroid of triangle

This formula is used to find the centroid of a triangle when the vertices of the traingle say (x1, y1) , (x2, y2) and (x3, y3) are given.

• $\sf Centroid = \left(\dfrac{x_1 + x_2 + x_3}{3} , \dfrac{y_1+y_2+y_3}{3}\right)$

Area of triangle

If we are given coordinates of three vertices of a triangle, we can calculate it's area by using the below formula:

• $\sf Area =\dfrac{1}{2} \left | \begin{array} {ccc} \sf{x_1}& \sf{y_1} &\sf{1} \\ \sf{x_2}& \sf{y_2}& \sf{1 }\\ \sf{x_3}& \sf{y_3} &\sf{1}\end{array} \right |$

Slope

Slope is the numeric description of the steepness of a line. It is denoted by 'm'. It can be calculated by two methods - when the inclination of the line with x axis measured anti-clockwise is given and when two coordinates through which the line passes is given.

When inclination is given

Let's say we are given that a straight line makes θ angle anti-clockwise with positive x axis then it's slope will be given by,

• $\sf m = tan \theta$

When two points are given

Let's say we are given two points through which the given line passes, let them (x1, y1) and (x2, y2), then the slope of this line will be given by,

• $\sf m = \dfrac{y_2-y_1}{x_2-x_1}$

Concept based on slope

• Slope of horizontal line is 0.
• Slope of vertical line is not defined.
• When two lines are perpendicular then the slope of both lines will be negative reciprocal of each other.
• When two lines are parallel, then the slope of both lines will be same.
• When three points are collinear, then the line joining any of the two pairs of coordinate will have the same slope.

Equations of line:-

There are several forms of equation of a straight line. Any line can be expressed in any of these forms. We will discuss all the forms one by one.

Slope point form

When we are given one point through which a line passes and it's slope is given, then the equation of a line is given by,

• $\sf (y-y_1) = m(x-x_1)$

Two point form

When we are given two points through which the given line passes, then the equation will be,

• $\sf (y-y_1) = \left(\dfrac{y_2-y_1}{x_2-x_1}\right)(x-x_1)$

Slope intercept form

When we are given slope and x or y intercept, we can express the equation of line as,

• $\sf y = mx + c$
• $\sf y = m(x-d)$

Here the first one is for y intercept which is 'c' and second equation is for x intercept which is 'd'.

Normal form

When we are given perpendicular distance from origin to the line and the angle made by the perpendicular with the positive x axis, we use this form of straight line.

• $\sf x \,cos\,\alpha + y\, sin\, \alpha = p$

Here alpha and p are the angle and perpendicular distance respectively.

Intercept form

When we are given both x as well as y intercept, we may use this form of straight line equation.

• $\sf \dfrac{x}{a} + \dfrac{y}{b} = 1$

Here a and b are x and y intercept respectively.

Angle between two lines

When we are given the slopes of two lines (or if that's possible to find using given equation maybe), then the acute angle between those lines will be given by,

• $\sf tan (\theta) = \left|\dfrac{m_1-m_2}{1+m_1.m_2}\right|$

Here theta is the acute angle between two given lines.

Distance of point and line

Whenever we are given an equation of a line say Ax + By + C = 0 and a point say (x1, y1) and are asked to find the perpendicular distance of the point from the line, we may use the following formula:

• $\sf d = \left|\dfrac{Ax_1 + By_1 + C}{\sqrt{A^2+B^2}}\right|$

Distance between parallel lines

Whenever we are equations of two lines which are parallel to each other, we may use the following formula to find the distance between them.

• $\sf d = \dfrac{|C_1-C_2|}{\sqrt{A^2+B^2}}$

Answer 2

Step-by-step explanation:

given :

• straight lines lesson all formulas

to find :

• lines lesson all formulas

solution :

What is a Straight Line?

• A line is simply an object in geometry that is characterized under zero width object that extends on both sides. A straight line is just a line with no curves. So, a line that extends to both sides till infinity and has no curves is called a straight line

Equation of Straight Line

• The relation between variables x, y satisfy all points on the curve.

• The general equation of straight line is as given below:

• ax + by + c = 0 { equation of straight line.

• Where x, y are variables and a,b, c are constants.

angle with + ve x-axis

'tan e' is called slope of straight line.

• The equation of the line with slope m and y-intercept c is y=mx+c,

Note 1 – If line is Horizontal, then slope = 0

Note 2 – If line is ⊥ to x-axis, i.e. vertical then slope is undefined.

Intercept Form:

• Equation of line with x – intercept as ‘a’ and y – intercept as ‘b’ can be given as

• x/a + y/b = 1

• x – co-ordinate of point of intersection of line with x-axis is called x-intercept.

• y – intercept will be y-co-ordinate of point of intersection of line with y-axis.

• For example,

• Along x-axis: x – Intercept = 5 and y – Intercept = 0

• Along y-axis: y – Intercept = 5 and x – Intercept = 0

• Also,

• Length of x – intercept = |x1|

• Length of y – intercept = |y1 |

• Note: Line passes through origin, intercept = 0

• x – Intercept = 0

• y – Intercept = 0

Again,

• ON = P

• AON = α

• Let length of \bot⊥ r from origin to S.L is ‘P’ and let this \bot⊥r make an angle with + vex- axis ‘α’, then equation of a line can be

• x cos a + y sin a = p x/ p sec a + y / pcos eca

• = y sin = p