Hello,

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Originally Posted by

**Hweengee** Got kind of stuck on this question.

Suppose a is a non-zero constant. x^2/3+y^2/3=a^2/3. Show that the length of the portion of any tangent line to this astroid cut off by the x and y axes is constant.

I've differentiated implicitly to obtain the gradient of the tangent as $\displaystyle dy/dx=-(y/x)^1/3.$ And have obtained the equation of the tangent line passing through (q,0) and (0,p) as y=-(y/x)^1/3.x+p.

There are many x's and y's in the equation of the line. (Wondering) Let $\displaystyle M(x_0,y_0)$ be a point of the astroid. The equation of the tangent line at $\displaystyle M$ is

$\displaystyle y=-\left( \frac{y_0}{x_0}\right)^\frac13(x-x_0)+y_0$

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But how do i show that the length of this line passing through (q,0) and (0,p) is constant?

Let $\displaystyle P(0,p)$ and $\displaystyle Q(0,q)$ be the intersection point of the tangent line with the y-axis and the x-axis, respectively. $\displaystyle p$ satisfies

$\displaystyle p=-\left( \frac{y_0}{x_0}\right)^\frac13(0-x_0)+y_0$

and $\displaystyle q$ is such that

$\displaystyle 0=-\left( \frac{y_0}{x_0}\right)^\frac13(q-x_0)+y_0$

You can now solve these two equations for $\displaystyle p$ and $\displaystyle q$ and compute $\displaystyle PQ=\sqrt{p^2+q^2}$ to check that this length is constant. (remember that we have $\displaystyle x_0^\frac23+y_0^\frac23=a^\frac23$)