Question

if tangent any point on the curve x⅔+y⅔=a⅔ intersects the coordinate axis In A and B then show that the length AB is constant​

Answer 1

Answer:

Step-by-step explanation:

Equation of the curve is x  

2/3

+y  

2/3

=a  

2/3

 

the parametric equations of the curve are x=acos  

3

θ,

y=asin  

3

θ

⇒  

dx

d

=  

dθ

dx

​  

 

dθ

dy

​  

 

​  

=  

a⋅3cos  

2

θ(−sinθ)

a⋅3sin  

2

θcosθ

​  

 

=−tanθ

Equation of the tangent at (acos  

3

θ,asin  

3

θ) is y−asin  

3

θ=−tanθ(x−acos  

2

θ)

y−asin  

3

θ=−xtanθ+a  

cosθ

sinθ

​  

cos  

3

θ

y−asin  

3

θ=−x  

cosθ

sinθ

​  

+asinθcos  

2

θ

⇒ycosθ+xsinθ=asinθcos  

3

θ+a  

cosθ

sin  

3

θ

​  

 

⇒xsinθ+ycosθ=asinθcosθ(cos  

2

θ+sin  

2

θ)

⇒  

cosθ

x

​  

+  

sinθ

y

​  

=a     ( Divided by both side sinθcosθ)

x-intercept =acosθ, y-intercept=asinθ

A(acosθ,0) and B(0,asinθ)

AB=  

(acosθ)  

2

+(asinθ)  

2

 

​  

 

=  

a  

2

(cos  

2

θ+sin  

2

θ)

​  

 

=  

a  

2

 

​  

=a

∴AB=a, a constant.