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Question: The degree of the differential equation of all tangent lines to the parabola $y^2=4ax$ is:...

The degree of the differential equation of all tangent lines to the parabola y2=4axy^2=4ax is:

A

1

B

2

C

3

D

None of these

Answer

2

Explanation

Solution

To find the degree of the differential equation of all tangent lines to the parabola y2=4axy^2=4ax, we first need to derive the differential equation.

The equation of a tangent to the parabola y2=4axy^2=4ax in terms of its slope 'm' is given by: y=mx+amy = mx + \frac{a}{m} ---(1)

This equation represents all tangent lines to the parabola and contains one arbitrary constant 'm'. To form the differential equation, we need to eliminate this arbitrary constant.

Differentiate equation (1) with respect to xx: dydx=m\frac{dy}{dx} = m ---(2)

Now, substitute the value of 'm' from equation (2) into equation (1): y=(dydx)x+a(dydx)y = \left(\frac{dy}{dx}\right)x + \frac{a}{\left(\frac{dy}{dx}\right)}

To clear the denominator, multiply the entire equation by dydx\frac{dy}{dx}: y(dydx)=x(dydx)2+ay \left(\frac{dy}{dx}\right) = x \left(\frac{dy}{dx}\right)^2 + a

Rearrange the terms to get the differential equation in a standard form: x(dydx)2−y(dydx)+a=0x \left(\frac{dy}{dx}\right)^2 - y \left(\frac{dy}{dx}\right) + a = 0

The degree of a differential equation is the power of the highest order derivative, when the differential equation is expressed as a polynomial in derivatives. In this differential equation, the highest order derivative is dydx\frac{dy}{dx} (which is a first-order derivative). The powers of dydx\frac{dy}{dx} present in the equation are 2 and 1. The highest power of the highest order derivative (dydx\frac{dy}{dx}) is 2.

Therefore, the degree of the differential equation is 2.