Draw 10 Concentric Circles in the Center of the Applet
In Geometry, the objects are said to be concentric when they share a common centre. Circles, spheres, regular polyhedra, regular polygons are concentric every bit they share the same centre bespeak. In Euclidean Geometry , ii concentric circles should accept different radii from each other. In this commodity, you will larn what are concentric circles, the theorem on concentric circles, the region betwixt the concentric circles, equations, and examples in detail.
Concentric Circles Meaning
The circles with a common centre are known as concentric circles and accept unlike radii. In other words, it is defined as two or more circles that have the same middle signal. The region betwixt two concentric circles are of unlike radii is known as an annulus.
Concentric Circumvolve Equations
Let the equation of the circumvolve with centre (-g, -f) and radius √[gii+f2-c] exist
ten2 + ytwo + 2gx + 2fy + c =0
Therefore, the equation of the circle concentric with the other circumvolve exist
xtwo + y2 + 2gx + 2fy + c' =0
It is observed that both the equations take the same middle (-one thousand, -f), but they have different radii, where c≠ c'
Similarly, a circle with heart (h, 1000), and the radius is equal to r, then the equation becomes
( x – h )2 + ( y – k )2 = r2
Therefore, the equation of a circle concentric with the circle is
( 10 – h )ii + ( y – k )2 = r1 ii
Where r ≠ r1
Past assigning unlike values to the radius in the above equation, we shall get a family of circles.
Concentric Circles – Theorem
In two concentric circles, the chord of the larger circle, which touches the smaller circle, is bisected at the signal of contact.
Proof
Given:
Consider ii concentric circles C1 and C2, with centre O and a chord AB of the larger circle C1, touching the smaller circle C2 at the point P as shown in the figure below.
Construction:
Join OP.
To show: AP = BP
Proof:
Since AB is the chord of larger circle Ci, it becomes the tangent to C2 at P.
OP is the radius of circle C2.
We know that the radius is perpendicular to the tangent at the point of contact.
So, OP ⊥ AB
Now AB is a chord of the circle C1 and OP ⊥ AB.
Therefore, OP is the bisector of the chord AB.
Thus, the perpendicular from the center bisects the chord, i.east., AP = BP.
Region Betwixt Concentric Circles
As mentioned higher up, the region betwixt two concentric circles is chosen the annulus. Nonetheless, we can observe the perimeter and area of the annulus using advisable formulas. The area of the annulus is calculated past subtracting the area of smaller circles from the area of the larger circle.
Suppose R is the radius of the larger circle and r is the radius of the smaller circle such that the area of the region bounded by these 2 circles is given by:
Area of annulus = πRii – πrtwo
Learn more about annulus here.
Concentric Circle Examples
Question: Find the equation of the circle concentric with the circle xtwo + ytwo + 4x – 8y – 6 =0, having the radius double of its radius.
Solution:
Given, circle equation: x2 + y2 + 4x – 8y – half dozen =0
We know that the equation of the circle is x2 + y2 + 2gx + 2fy + c =0
From the given equation, the center point is (-two, 4)
Therefore, the radius of the given equation will be
r = √[one thousandtwo+ftwo-c]
r = √[iv+16+6]
r = √26
Permit R be the radius of the concentric circle.
It is given that, the radius of the concentric circumvolve is double of its radius, then
R = 2r
R = 2√26
Therefore, the equation of the concentric circle with the radius R and the heart bespeak (-m, -f ) is
( x – k )2 + ( y – f )2 = Rtwo
(ten + 2)two + ( y – four )2 = (2√26 )2
tenii + 4x + 4 + y2 – 8y + sixteen = four (26)
x2 + yii + 4x – 8y +twenty = 104
xtwo + y2 + 4x – 8y – 84 = 0
Question ii:
Find the area between ii concentric circles whose diameters are 35 cm and 21 cm.
Solution:
Given,
The diameters of the 2 circles are 35 cm and 21 cm.
Then, R = 35/ii cm
r = 21/2 cm
Area of betwixt concentric circles = πR2 – πr2
= (22/7) × (35/2) × (35/2) – (22/2) × (21/2) × (21/2)
= (22/7)[(35/ii) × (35/2) – (21/2) × (21/2)]
= (22/7) × [(352 – 212)/4]
= (22/7) × (56 × 14)/4
= 616 cm2
Stay tuned with BYJU'Due south – The Learning App to learn Maths-related concepts, and also watch engaging videos to learn with ease.
mcclearyroonstank.blogspot.com
Source: https://byjus.com/maths/concentric-circles/
0 Response to "Draw 10 Concentric Circles in the Center of the Applet"
Post a Comment