in the figure, O is the centre of the circle and X square+ Y square=25.(a) what is the radius of the circle?(b) write the equation of the circle who's centre is at orgin and radius 3?
Answers
Step-by-step explanation:
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Correct option is A)
Let the radius of the circle be 'R'
Let the rectangle inscribed in the circle be ABCD
If AB = l and BC = b
then Area of rectangle = lb,
Area of circle = πR²
Given that Ratio of Area of circle to Area of Rectangle
= π : √3
⇒
lb
πR
²
=
√3
π
R²
lb
=√3
let x=
R
l
andy=
R
b
, then
xy = √3
Consider triangle ABC, which is right angled at B, since ABCD is
a rectangle, so
AC² = AB² + BC²
But AC is the chord of the circle subtending 90° at B point on
circle, so AC should be diameter and AC = 2R
4R² = l² + b²
(
R
l
)²+(
R
b
)²=4
⇒ x² + y² = 4
But we know xy = √3
So x²y² = 3
x² + 3/x² = 4
x⁴ - 4x² + 3 = 0
So x² = 1 or x² = 3
⇒ x = 1 or √3
If x = 1, then y = √3 or vice-versa
If we assume x > y, then x : y = √3 : 1
But x : y = l : b = √3 : 1
Consider Δ ADE and ΔBDC ,
∠DAE = ∠DCB = 90°
∠ADE = ∠BDC = ∠ODC (Given)
Hence , ΔADE ≈ ΔBDC
Since both triangles are similar, ratio of the sides corresponding
to equal angles would be equal
BC
AE
=
DC
AD
⇒
AD
AE
=
DC
BC
=
√3
1
(Assuming BC < DC)
or √3 (if BC > DC)