#### To determine

**To show:**

The length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant

#### Answer

The length of the portion of any tangent line to the asteroid cut off by the coordinate axes is constant

#### Explanation

**1.Formula:**

(i) Chain rule:

dydx= dydududx

(ii) Power Rule:

ddxxn=n xn-1

(iii) Standard Formula:

ddxconstant=0

(iv) Slope-point form:

y-y1=m x-x1

(v) Use distributive property:

a (b+ c) = a*b + a*c

**2. Given:**

x2/3+y2/3=a2/3

**3.Calculation:**

Differentiating implicitly y with respect to x,

By using power rule

23x23-1+23y23-1dydx=0

23x-13+23y-13dydx=0

Subtracting

23x-13

from both sides

23y-13dydx=-23x-13

Cancel out the common terms

y-13dydx=-x-13

dydx=-x-13y-13

dydx=-yx1/3

dydx=slope=m=chnage in ychnage in x

∴m=-yx1/3

Equation of tangent line passing through point (x0,y0) on curve is,

y-y0=-y0x01/3x-x0

The tangent line intercepts the x axis at y=0 and at y axis at x=0

Now solving for x intercept, substitute y=0

0-y0=-y0x01/3x-x0

-y0=-y0x01/3x-x0

Solving for x,

y0=y0x01/3x-x0

By using distributive property

y0=y0x01/3x-y0x01/3x0

y0x01/3x=y0+y0x01/3x0

Dividing the whole equation by y0x01/3

x=y0+y0x01/3x0y0x013

x=y0x0y01/3+x0

x=y01-13x01/3+x0

x=y023x01/3+x0

Now, (x0,y0) satisfies asteroid equation

x02/3+y02/3=a02/3

Solving for y02/3

y02/3=a02/3-x02/3

x=a02/3-x02/3x01/3+x0

By using Distributive Property

x=a02/3 x01/3-x02/3 x01/3 +x0

x=a02/3 x01/3-x0 +x0

x=a02/3 x01/3

Similarly, solving for y intercept, substitute x=0

y-y0=-y0x01/3x-x0

y-y0=-y0x01/30-x0

y-y0=-y0x01/3-x0

y-y0=y0x01/3x0

y-y0=y01/3 x0-13x0

y-y0=y01/3x023

Solving for y

y=y01/3x023+y0

y=y01/3a02/3-y02/3+y0

By using Distributive Property

y=y01/3a02/3-y01/3 y02/3+y0

y=y01/3a02/3-y0+y0

y=y01/3a02/3

Now squaring and adding x and y intercept, we get the distance portion of any tangent line to the astroid cut off by the coordinate axes. Thus we have distance,

=a02/3 x01/32+y01/3a02/32

=a04/3 x02/3+ y02/3a04/3

Taking a04/3 common,

=a04/3 x02/3+ y02/3

=a04/3 a02/3

=a02=constant

Since our choice of tangent line was arbitrary this is true for all tangent lines.

**Conclusion:**

Therefore, the length of the portion of any tangent line to the astroid cut off by the coordinate axes is constant