P(x, y)

Dividing Point of a Line Segment

Learn what a point is, see how it’s defined, then watch the equation of the circle get derived — one step at a time.

Straight Line

When the equation 3x − 2y − 4 = 0 is represented on a graph, it is observed that an equation of degree 1 represents a straight line.

The equation of a straight line is also called a linear equation.

In general, straight lines are of three types: Lines parallel to the X-axis, lines parallel to the Y-axis, and lines making an angle with the X-axis.

These can be seen in the figures given below:

Lines parallel to the X- axis,Horizontal straight line
Lines parallel to the Y- axisVertical straight line
Making the angle tothe X- axis
Student Concept Breakdown & Assistant Guide

1. Horizontal Lines (AB & CD)

Parallel to the X-axis. Every point on the line keeps the exact same y-value.

Form: y = k (slope m = 0)

2. Vertical Lines (RS & PQ)

Parallel to the Y-axis. Every point on the line keeps the exact same x-value.

Form: x = k (slope m = undefined)

3. Slanted / Inclined Lines (LM)

Cuts through both axes making angle θ with the positive X-axis.

Form: y = mx + c or ax + by + c = 0
Click on any figure above to highlight its specific line type on the graph!
Equation Example: 3x - 2y - 4 = 0 → Degree = 1

Section Formula – Coordinates of a Point Dividing a Line Segment

Step 1 of 7
Y Y' X X' O A(x₁, y₁) P(x, y) B(x₂, y₂) M N D F E

Section Formula

Let, in the figure, A(x1, y1) and B(x2, y2) are two points. P(x, y) divides the line segment AB internally in the ratio m1:m2. Calculate the coordinate of P(x, y).

Now, draw AD⊥OX, BE⊥OX, PF⊥OX, AM⊥PF and PN⊥BE

In the figure, AM = DF = OF − OD = x − x1,

PN = FE = OE − OF = x2 − x

PM = PF − MF = PF − AD = y − y1

BN = BE − NE = BE − PF = y2 − y

And APBP = m1m2

Right angled triangles ΔPMA and ΔBNP are similar.

AMPN = APBP [ ∵ The ratio of corresponding sides of similar triangle]

Or, x − x1x2 − x = m1m2

Or, m2x − m2x1 = m1x2 − m1x

Or, m2x + m1x = m1x2 + m2x1

Or, x(m1 + m2) = m1x2 + m2x1

Or, x = m1x2 + m2 x1m1 + m2

Again, PMBN = APBP [ ∵ The ratio of corresponding sides of similar triangle]

Or, y − y1y2 − y = m1m2

Or, m2y − m2y1 = m1y2 − m1y

Or, m2y + m1y = m1y2 + m2y1

Or, y(m1 + m2) = m1y2 + m2y1

Or, y = m1y2 + m2y1m1 + m2

∴ (x, y) = (m1x2 + m2 x1m1 + m2 , m1y2 + m2 y1m1 + m2)

Thus, the coordinates of the point dividing line segment AB are

P(x, y) = (m1x2 + m2 x1m1 + m2 , m1y2 + m2 y1m1 + m2)

Slope of the Line Joining the Points (x1, y1) and (x2, y2)

Step 1 of 5
Y Y′ X X′ O A B P(x1, y1) Q(x2, y2) θ M N θ

Slope of the Line Joining Two Points

Let the line PQ, which passes through the points P (x1, y1) and Q (x2, y2) intersect the X- axis at point A and form an angle of ∠PAX = θ in the positive direction with the X- axis.

In the figure, draw PM⊥OX, QN⊥OX and PR⊥QN

Then, PR = MN = ON − OM = x2 − x1 and

QR = QN − RN = QN − PM = y2 − y1

with ∠QPR = ∠PAX = θ

In the right-angled triangle ΔPRQ,

∴ The inclination (or slope) of line PQ(m) = tanθ

= QRPR

= y2 − y1x2 − x1

Calculation of x- intercept and y- intercept

Step 1 of 5
Y Y′ X X′ O B(0, b) A(a, 0) b a

Calculation of x- intercept and y- intercept

How do you find the x- intercept and y- intercept of a straight line? Let's discuss.

In the adjacent figure, the straight line AB intersects the X- axis at point A(a, 0) and the Y- axis at point B(0, b).

This means that OA = a and OB = b.

These segments are respectively called the x- intercept and the y- intercept.

∴ x- intercept = a, y- intercept = b.

Slope or Gradient of a Straight Line

Step 1 of 3
Y X Wall O A Ground B Ladder y – intercept X – intercept tanθ = slope

Slope or Gradient of a Straight Line

The ladder must have been placed leaning against the wall, tilted or inclined. The extent to which the ladder is tilted or inclined while placing it is called the inclination of the ladder with the ground.

OA = the part made by line AB with the X-axis, called the x- intercept, and

OB = the part made by line AB with the Y-axis, called the y- intercept.

The angle θ that the ladder makes with the ground (the X-axis) is its inclination.

tanθ = slope

Thus, the tangent of the angle of inclination is called the slope or gradient of the line.

Example 1 of 2

Solve It