Find the equation of the normal to the curve x+y=x^y, where it cuts the x-axis.

Name: Problem 10, Chapter 8: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-02-27T15:53:51-08:00
Duration: 1 min 18 s
Channel: Varsha Aggarwal
Description: Find the equation of the normal to the curve x+y=x^y, where it cuts the x-axis.

step 1 So equation of curve is x plus y equal x to the power y. Now put y equal 0, then x equal 1. So p

Find the equation of the normal to the curve x+y=x^y, where...

Key Concepts

Point-Slope Form

The point-slope form is a way of expressing the equation of a line when a point on the line and its slope are known. It is given by the formula y - y? = m(x - x?), where (x?, y?) is the point and m is the slope. This form is particularly useful when deriving the equation of the tangent or normal line.

Implicit Differentiation

Implicit differentiation is a technique used when a function is defined implicitly rather than explicitly. Instead of solving for one variable in terms of the other, we differentiate both sides of the equation with respect to the independent variable, applying the chain rule whenever necessary. This method is particularly useful for curves that cannot be easily rearranged into y = f(x).

Logarithmic Differentiation

Logarithmic differentiation is a method that involves taking the natural logarithm of both sides of an equation before differentiating. This technique is especially useful for differentiating functions where variables appear in both the base and the exponent. The logarithm helps convert multiplicative relationships into additive ones, simplifying the differentiation process.

Tangent Line

The tangent line to a curve at a given point represents the instantaneous direction of the curve at that point. Its slope is given by the derivative of the function at that point. Finding the equation of the tangent line is often the first step in determining related concepts, such as the normal line.

Normal Line

The normal line to a curve at a given point is the line perpendicular to the tangent line at that point. Since the slopes of perpendicular lines are negative reciprocals of each other, once the tangent slope is determined, the normal slope can be calculated, which is fundamental in forming the equation of the normal.

x-Intercept Calculation

Determining the x-intercept of a line involves finding the point where the line crosses the x-axis, which means setting y to 0 in the line’s equation. This process is a straightforward application of algebra and is crucial when the problem requires identifying specific intercepts of lines.

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