If the line ax+by+c=0 is a normal to the curve xy=1, then
If the line ax+by+c=0 is a normal to the curve xy=1, then
a>0, b>0
a>0, b<0
a<0,b<0
a=0, b=0
The Correct Option is B
Solution and Explanation
The curve xy = 1 can be written as y = 1/x, which means that the derivative of y with respect to x is: dy/dx = -1/x^2
For a normal to the curve at a given point, the slope of the tangent at that point is given by: m = -1/dy/dx = x^2
Therefore, the equation of the tangent at the point (a, 1/a) is: y - 1/a = x^2 (x - a)
Simplifying, we get: y = a^2 x + (1 - a^3)/a This is the equation of the tangent line.
For this line to be a normal to the curve xy = 1, it must be perpendicular to the curve at the point (a, 1/a).
The slope of the curve at this point is: dy/dx = -1/x^2 = -a^2
Therefore, the slope of the line perpendicular to the curve is: m = 1/a^2
This means that the product of the slopes of the tangent and the normal at the point (a, 1/a) is: m * (-a^2) = -1
Solving for a, we get: a = ±1 Substituting a = ±1 in the equation of the tangent line, we get: y = ±x + 1
These are the equations of the two lines that are normal to the curve at the points (1, 1) and (-1, -1).
The normal at (1, 1) has a positive slope, and the normal at (-1, -1) has a negative slope. Therefore, the correct option is (B) a > 0, b < 0.
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≤ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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