top button
Flag Notify
    Connect to us
      Facebook Login
      Site Registration Why to Join

    Get Free Puzzle Updates

Facebook Login
Site Registration

A circle of radius 1 is tangent to the parabola y=x^2 as shown. Find the gray area between the circle and the parabola?

+1 vote
58 views

A circle of radius 1 is tangent to the parabola y=x^2 as shown. Find the gray area between the circle and the parabola?

enter image description here

posted May 25 by anonymous

Share this puzzle
Facebook Share Button Twitter Share Button Google+ Share Button LinkedIn Share Button Multiple Social Share Button

1 Answer

0 votes

Equation of the circle = x^2 + (y - k)^2 = 1 ----- c {k is the 'y' coordinate of the centre of the circle}
Equation of the parabola y = x^2 ----- p
at the point of contact, the slopes of both the equations must be the same.

y=x^2
(dy/dx) = 2x {Parabola} ---- 1

y = -(1 - x^2)^0.5 + k (we just need the lower part of the circle to calculate the area}
(dy/dx) = -(1/(2(1 - x^2)^0.5))*(-2x) = x/(1 - x^2)^0.5 {Circle} ------ 2

1 = 2

2x = x/(1 - x^2)^0.5
(1 - x^2)^0.5 = 1/2
1 - x^2 = 1/4
x^2 = 3/4
x = 3^0.5/2 ---- 3

Sub 3 in p
y = x^2 = 3/4 = 0.75 ---- 4

Sub 4 and 3 in y = -(1 - x^2)^0.5 + k
0.75 + (1 - 3/4)^0.5 = k
k = 1.25

Therefore the coordinates of the intersection are (3^0.5/2, 0.75) & (-3^0.5/2, 0.75).

Required area = integration of ( -(1 - x^2)^0.5 + 1.25 - x^2) from x = -3^0.5/2 to x = 3^0.5/2. = 0.251841 Sq units.

answer May 26 by Tejas Naik



Similar Puzzles
+2 votes

In a circle of radius 1, an equilateral triangle is inscribed in the circle as shown. What is the area of the blue region?

What is the area of the blue region

+1 vote

In the image below there are two tangent semicircles. The side length of the square is 36cm.
Radius of Yellow Circle

What is the length of the radius of the yellow circle?

0 votes

Seven circles, each with radius 1, overlap to form the composite figure shown below. Find the area of shaded region?

Find the area of the shaded region

+1 vote

7 non-overlapping unit circles are inscribed inside a large circle as shown below. Find the area of the shaded region?

enter image description here

Contact Us
+91 9880187415
sales@queryhome.net
support@queryhome.net
#280, 3rd floor, 5th Main
6th Sector, HSR Layout
Bangalore-560102
Karnataka INDIA.
QUERY HOME
...