Two-point Form of a Line

We will discuss here about the method of finding the equation of a straight line in the two point form.

To find the equation of a straight line in the two point form,

Let AB be a line passing through two points A (x$_{1}$, y$_{1}$) and B (x$_{2}$, y$_{2}$).

Let the equation of the line be y = mx + c ................... (i), where m is the slope of the line and c is the y-intercept.

As (x$_{1}$, y$_{1}$) and (x$_{2}$, y$_{2}$) are points on the line AB, (x$_{1}$, y$_{1}$) and (x$_{2}$, y$_{2}$) satisfy (i).

Therefore, y$_{1}$ = mx$_{1}$ + c ................................ (ii)

and y$_{2}$ = mx$_{2}$ + c ................................ (iii)

Subtracting (iii) from (ii),

y$_{1}$ - y$_{2}$ = m(x$_{1}$ - x$_{2}$)

⟹ m = $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ ................................ (iv)

Substituting m = $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ in (ii),

y$_{1}$ = [$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$]x$_{1}$ + c

⟹ c = y$_{1}$ - $\frac{x_{1}(y_{1} - y_{2})}{ x_{1} - x_{2}}$

⟹ c = $\frac{ y_{1}(x_{1} - x_{2}) - x_{1}(y_{1} - y_{2})}{ x_{1} - x_{2}}$

⟹ c = $\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$

Therefore, from (i),

y = [$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$]x + $\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$

Subtracting y$_{1}$ from both sides of (v)

y - y$_{1}$ = [$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$]x + $\frac{x_{1}y_{2} - x_{2}y_{1}}{ x_{1} - x_{2}}$

⟹ y - y$_{1}$ = [$\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$]x + $\frac{x_{1}(y_{2} - y_{1})}{ x_{1} - x_{2}}$

⟹ y - y$_{1}$ = $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$(x + x$_{1}$)

The equation of the straight line passing through (x1, y1) and (x2, y2) is y - y$_{1}$ = $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$(x + x$_{1}$)

Note: From (iv), the slope of the line joining the points (x$_{1}$, y$_{1}$) and (x$_{2}$, y$_{2}$) is $\frac{y_{1} - y_{2}}{x_{1} - x_{2}}$ i.e., $\frac{Difference of y-coordinates}{difference of x-coordinates in the same order}$

Solved example on two-point form of a line:

The equation of the line passing through the points (1, 1) and (-3, 2) is

y - 1 = $\frac{1 - 2}{1 - (-3)}$(x - 1)

⟹ y – 1 = -$\frac{1}{4}$(x – 1)

Also, y – 2 = $\frac{2 - 1}{-3 - 1}$(x + 3)

⟹ y – 2 = -$\frac{1}{4}$(x + 3)

However, the two equations are the same.

● Equation of a Straight Line

• Inclination of a Line
• Slope of a Line
• Intercepts Made by a Straight Line on Axes
• Slope of the Line Joining Two Points
• Equation of a Straight Line
• Point-slope Form of a Line
• Two-point Form of a Line
• Equally Inclined Lines
• Slope and Y-intercept of a Line
• Condition of Perpendicularity of Two Straight Lines
• Condition of parallelism
• Problems on Condition of Perpendicularity
• Worksheet on Slope and Intercepts
• Worksheet on Slope Intercept Form
• Worksheet on Two-point Form
• Worksheet on Point-slope Form
• Worksheet on Collinearity of 3 Points
• Worksheet on Equation of a Straight Line

10th Grade Math

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