Equation of a Straight Line, passes through Origin with slope

Expressing a straight line in algebraic form expression when the straight line passes through the origin with some slope is defined equation of a straight line, passes through the origin with slope.

In very special cases, straight lines are intersected the origin of the Cartesian coordinate system with some slope and these types of straight lines can be expressed in terms of an algebraic expression. It is very easier to represent it in mathematics by a single expression which reveals how exactly the straight line is in the geometric system. Actually, the straight line is expressed in terms of the slope and any one of the points on the straight line.

Geometric Explanation

Assume, AB is a straight line on a plane and also assume it intersects the origin of the Cartesian coordinate geometric system with some slope.

Straight line passing through the origin

The straight line AB intersects the origin at a point A which is origin as well as one of the points of the straight line AB. Therefore, the coordinates of the point A is (0, 0) geometrically. Assume, the point B is located at (x, y) in the Cartesian coordinate geometric system. The assumed angle made by the straight line AB with the horizontal x-axis is theta (θ).

Straight line passing through the origin forms a right angled triangle

Draw a perpendicular line from point B towards the x-axis and it intersects the horizontal x-axis at a point which is assumed to call point C. Thus, it forms a right angled triangle ΔBAC. The angle made the straight line AB with horizontal x-axis is theta (θ) but it is also an angle of the right angled triangle ΔBAC which means BAC = θ.

As per this right angled triangle,

  • The line segment AB becomes the hypotenuse of the right angled triangle ΔBAC.
  • The line segment AC becomes the adjacent side of the right angled triangle ΔBAC.
  • The line segment BC becomes the opposite side of the right angled triangle ΔBAC.

As per the principle definition of slope of the straight line, the slope of the straight line is mathematically expressed as m = tanθ .

But, as per the right angled triangle ΔBAC, tanθ = BCAC

As per the right angled triangle ΔBAC,

The length of the opposite side (BC) of the right angled triangle ΔBAC is BC = y

The length of the adjacent side (AC) of the right angled triangle ΔBAC is AC = x

Therefore, tanθ = BCAC = yx

tanθ = yx

Replace the value of tanθ in slope form.

m = yx

mx = y

y = mx

This is an algebraic expression which represents a straight line when the straight line passes through the origin of the Cartesian coordinate geometric system.

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