Find the equations of all lines having slope 0 which are tangent to the curve y=(1)/(x^2-2 x+3).

Name: Problem 12, Chapter 6: NCERT Class 12 Part 1 - Math
Uploaded: 2021-06-22T02:15:26-08:00
Duration: 1 min 6 s
Channel: Tanishq Gupta
Description: Find the equations of all lines having slope 0 which are tangent to the curve y=(1)/(x^2-2 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 equations of all lines having slope 0 which are...

Key Concepts

Tangent Lines

A tangent line to a curve is a line that touches the curve at a specific point without crossing it. In calculus, the tangent line represents the best linear approximation of the curve at that point and its slope is given by the derivative of the curve’s function at that point.

Horizontal Tangents

Horizontal tangents are tangent lines with a slope of zero. They occur at points on the curve where the derivative is zero, meaning the instantaneous rate of change is zero. Identifying horizontal tangents is essential when studying the local behavior of functions, such as determining local maxima, minima or points of inflection.

Tangency Condition via Discriminant

For a line to be tangent to a curve, when setting the line and curve equations equal, the resulting equation must have exactly one solution (a repeated root). In the context of quadratic equations, this happens when the discriminant (the part under the square root in the quadratic formula) is zero. This condition ensures that the line touches the curve at one distinct point.

Algebraic Manipulation of Rational Functions

When dealing with rational functions, such as the given curve, it is often necessary to manipulate the function algebraically to identify its behavior and simplify the process of finding tangent lines. Techniques such as simplifying denominators or completing the square can be important tools in setting up the equation whose roots determine the points of tangency.

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