Notes On Equations of Lines (Part I) - CBSE Class 11 Maths

If a point lies on a line, then it must satisfy the equation of the line.

Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line L at any position is 'a'.

Equation of line l: y = a


Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line S at any position is minus 'a'.

Equation of line s: y = -a

Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis.

The x coordinate of every point on line L at any position is 'b'.

Equation of line l: x = b

Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis.

The x coordinate of every point on line S at any position is minus 'c'.


Equation of line s: y = -c

Point-Slope Form

Consider a non-vertical line L passing through a given point P(x1, y1).

Let m be the slope of line L.

Let A(x, y) be any arbitrary point on line L.

The slope of a line passing through (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1).

Here, slope (m) = (y - y1)/(x - x1)

y - y1 = m (x - x1) ….(1)


Equation 1 represents the point-slope form of the equation of a line passing through a given point (x1, y1) and with slope m.

Two-Point Form

Consider a non-vertical line L passing through two given points P and Q.

Let A be any arbitrary point on line L.

Three given points A, B and C are collinear if: Slope of AB = Slope of BC

Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ 

Slope of PA = (y - y1)/ (x - x1)

Slope of PQ = (y2 - y1)/ (x2 - x1)

⇒ (y - y1)/ (x - x1) = (y2 - y1)/ (x2 - x1)

This equation represents the two-point form of the equation of a line passing through the given points, P and Q.

Summary

If a point lies on a line, then it must satisfy the equation of the line.

Consider a horizontal line L that intersects the positive side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line L at any position is 'a'.

Equation of line l: y = a


Consider another horizontal line S that intersects the negative side of the Y-axis at a distance 'a' from the X-axis.

The y coordinate of every point on line S at any position is minus 'a'.

Equation of line s: y = -a

Consider a vertical line L that intersects the positive side of the X-axis at a distance 'b' from the Y-axis.

The x coordinate of every point on line L at any position is 'b'.

Equation of line l: x = b

Consider another vertical line S that intersects the negative side of the X-axis at a distance 'c' from the Y-axis.

The x coordinate of every point on line S at any position is minus 'c'.


Equation of line s: y = -c

Point-Slope Form

Consider a non-vertical line L passing through a given point P(x1, y1).

Let m be the slope of line L.

Let A(x, y) be any arbitrary point on line L.

The slope of a line passing through (x1, y1) and (x2, y2) is (y2 - y1)/(x2 - x1).

Here, slope (m) = (y - y1)/(x - x1)

y - y1 = m (x - x1) ….(1)


Equation 1 represents the point-slope form of the equation of a line passing through a given point (x1, y1) and with slope m.

Two-Point Form

Consider a non-vertical line L passing through two given points P and Q.

Let A be any arbitrary point on line L.

Three given points A, B and C are collinear if: Slope of AB = Slope of BC

Thus, A, P and Q are collinear if: Slope of PA = Slope of PQ 

Slope of PA = (y - y1)/ (x - x1)

Slope of PQ = (y2 - y1)/ (x2 - x1)

⇒ (y - y1)/ (x - x1) = (y2 - y1)/ (x2 - x1)

This equation represents the two-point form of the equation of a line passing through the given points, P and Q.

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