Q:

How Do You Find the Equation of a Secant Line?

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Quick Answer

A secant line makes an intersection on a curve at two or more points, according to Khan Academy. Three things can happen when a line is drawn on a graph: The line may not intersect the curve, the line may intersect the curve at one point or the line may intersect the curve at more than one point. Anywhere with a curve and a line intersecting two or more points has a secant line. For example, arched bridges and bicycle wheels have secant lines. In an arched bridge, the arch is the curve and the road is a secant line. The arch in many arched bridges intersects the road in two or more places. Similarly, in a bicycle wheel, the wheel is the curve and the spokes are secant lines. The spokes of a bicycle wheel often intersect the wheel in two or more different places.

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The Calculation
In math, when two points intersect — call them (a1, b1) and (a2, b2) — it's possible to calculate the slope of a line at the two points of intersection via the formula (b2 - b1) / (a2 - a1). This formula describes the rate of change of one point of the intersection with respect to the other.

Once the slope of the line between the two points has been determined, it's possible to find the equation of the line through the points using the equation b - b1 = m(a - a1) where m is the slope, states Brightstorm. The results of the equation provide the slope of the line at a given point. This means that it's possible to find the equation of a line once the two points on a secant line are established.

To determine the slope of a line at a given point, one must first find two points of the secant line, and then find the slope of the line between the two points. One of the points and the slope are then used to determine the equation of the line.

Secant Lines, Tangent Lines and Limit Definition of a Derivative
Secant lines are simply two lines that join two points on a function. These lines are equivalent to the slope linking the points. On the other hand, for a line to qualify as a tangent line, it must be a straight line, and it must touch the function at a single point. A tangent line is a representation of immediate rate of variation of a given function at one point. The slope of a tangent line at a given point is the same as the derivative of the function at an equivalent point.

It's easy to see what happens when the distance between the two points of the secant line is reduced. As the distance between the two points of a secant line approaches zero, the average rate of change equals the instantaneous rate of change. This means that the secant line become the tangent line.

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