Final answer:
To show that the normal at any point θ to the curve is at a constant distance from the origin, we need to find the equation of the normal and then show that its distance from the origin remains constant for any value of θ.
Step-by-step explanation:
To show that the normal at any point θ to the curve is at a constant distance from the origin, we need to find the equation of the normal and then show that its distance from the origin remains constant for any value of θ.
The equation of the curve is given by x = acosθ + aθsinθ and y = asinθ - aθcosθ. To find the equation of the normal, we differentiate the equation of the curve with respect to θ to get the slope of the curve at any point. The slope of the normal will be the negative reciprocal of the slope of the curve. With the slope of the normal and a point on the curve, we can find the equation of the normal using the point-slope form.
Once we have the equation of the normal, we can find its distance from the origin using the distance formula. By substituting different values of θ, we can show that the distance remains constant.