# Differential and Integral Calculus Questions and Answers – Evolutes

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This set of Differential and Integral Calculus Problems focuses on “Evolutes”.

1. The name of the evolute of an ellipse is ___________
a) Centroid
b) Astroid
c) Asteroid
d) Cycloid

Explanation: Astroid is known as the evolute of ellipse whereas centroid is associated with triangle. Asteroids are small planet like structures found in space. Cycloid is the curve traced by a point on the circumference of a circle.

2. The curvature of a curve is equal to ____________
a) Reciprocal of radius of curvature
c) Twice the radius of curvature
d) One

Explanation: Curvature is the property of a curve by which the curve deviates from that of a straight line. It is equal to the reciprocal of the radius of curvature of the curve. The radius of curvature is equal to the radius of the curve by magnitude.

3. Involute is also known as ___________
a) Evolute
b) Evolvent
c) Envelope
d) Tangent

Explanation: The curve itself is the involute of the evolute of the curve with a known starting point. The other name for involute is evolvent.
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4. What is the evolute of parabola called?
a) Cycloid
b) Spiral
c) Congruent parabola
d) Semicubical parabola

Explanation: The evolute of parabola is called semicubical parabola. It is defined parametrically as
x=t2 and y=at3.

5. Definition of evolute of a curve is ___________
a) locus of centre of the given curve
b) locus of centre of tangential curve
c) locus of circumferential point on the curve
d) locus of tangent to the curve

Explanation: The evolute of a curve is defined as the locus of centre of the given curve. It is the path traced by the centre of the curve.

6. What is the curvature of the curve x2 + y2 = 25?
a) 5
b) 25
c) 0.5
d) 0.2

Explanation: From the equation, it is clear that the radius of the curve (circle) is 5 units. The curvature is equal to the reciprocal of radius. Hence, curvature = 1/5 = 0.2.

7. What is the curvature of straight line?
a) infinity
b) one
c) zero
d) length of the straight line

Explanation: A straight line is a curve of infinite radius. Curvature = 1/radius. Hence, the curvature of straight line is 1/infinity which is equal to zero.

8. Number of possible evolutes for a curve is ____________
a) Two
c) One
d) Infinity

Explanation: A curve possesses only a single evolute. It can have infinite number of involutes.

9. The curvature of a plane curve at K is _________
a) one
b) d ψ/ds
c) zero
d) infinity

Explanation: The curvature of a plane curve is given as d ψ/ds. The curvature of straight line is zero.

10. What is the radius of curvature of the curve xy = c2 at (c,c)?
a) c
b) 2
c) 2c
d) √2 c

Explanation: y = c2/x
Integrating,
$$y_1 = \frac{-c^2}{x^2} = \frac{-c^2}{c^2} = -1$$
$$y_2 = \frac{2c^2}{x^3} = \frac{2c^2}{c^3} = \frac{2}{c}$$
Radius of curvature = $$\frac{\left(1+y_{1}^{2}\right)^\frac{3}{2}}{y_2} = \frac{\left(1+1\right)^\frac{3}{2}}{\frac{2}{c}} = \frac{c.2^{\frac{3}{2}}}{2} = \sqrt{2}c.$$

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