Find the equation of all lines having slope 2 which are tangents to the curve y=(1)/(x-3), x ≠3

step 1 The question says that we have to find the equation of all lines having slope 2, which are tange

Find the equation of all lines having slope 2 which are...

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Key Concepts

Slope-Intercept Form

The slope-intercept form, given by y = mx + b, is a standard way to represent the equation of a line where m is the slope and b is the y-intercept. Using this framework allows one to incorporate given slope values directly and is useful for formulating the equation of a tangent line with a specific slope.

Tangency Condition

The tangency condition requires that at the point where a line is tangent to a curve, both the function and the line have the same y-value and the same slope. This condition typically leads to a system of equations that must be solved simultaneously to find the exact point(s) of tangency, serving as a critical step in problems involving tangent lines.

Tangent Lines

A tangent line to a curve is a line that touches the curve at exactly one point without crossing it at that location. At the point of tangency, the curve and the line share the same direction, meaning their slopes are equal. This concept is central in calculus for approximating and understanding the local behavior of functions.

Derivative

The derivative of a function at a given point provides the slope of the tangent line to the curve at that point. It is a fundamental concept in calculus used to analyze how functions change and to determine instantaneous rates of change, making it essential in problems involving tangents.

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