Answer to Question #306576 in Geometry for Anurag

Question

Trace the following curves:

i) 𝑥(4𝑎

2 + 𝑦

2

) = 8𝑎

3

ii) 𝑦

2

(𝑥

2 + 𝑦

2

) + 𝑎

2

(𝑥

2 − 𝑦

2

) = 0

iii) 𝑎

2𝑦

2 = 𝑥

2

(𝑎

2 − 𝑥

2

)

iv) 𝑥 = 𝑡

2

, 𝑦 = 𝑡 −

𝑡

3

3

v) 𝑟 = 𝑎(1 + 𝑠𝑖𝑛𝜃)

vi) 𝑟

2 = 𝑎

2

𝑠𝑖𝑛2𝜃

vii) 𝑟 = 𝑎 𝑐𝑜𝑠3𝜃

viii) 𝑟 = 𝑎(1 + 𝑠𝑖𝑛𝜃)

Answer 1

A )4a²sq. units

B) 158a² sq. units

C

9
16a2​ sq. units

D

none of these

Correct option is B)

ay²=x²(a−x)⇒y=± xa
a−x​Curve tracing : y=xaa−x​We must have x≤aFor 0<x≤a, y>0 and for x<0,y<0Also y=0⇒x=0,aCurve is symmetrical about x-axis.When x→−∞,y→−∞Also, it can be verified that y has only one point of maxima for 0<x<a.Area =2∫0a​xaa−x​dxaa−x​=t⇒−ax=t²⇒x=a(1−t2)⇒A=2∫10a(1−t²)t(−2at)dt=41​(t²−t⁴)dt=4a²[3t³−5t⁵]01=4a²1]=158a²sq. units