<s>[INST] Prove that the length of segment on tangent is constant for $y=\frac a2\ln{\frac{a+\sqrt{a^2-x^2}}{a-\sqrt{a^2-x^2}}}-\sqrt{a^2-x^2}$ Prove that the length of segment of tangent from point of tangency to the point where it cuts the y-axis is constant. 
$$y=\frac a2\ln{\frac{a+\sqrt{a^2-x^2}}{a-\sqrt{a^2-x^2}}}-\sqrt{a^2-x^2}$$

After diffrentiating:
$$y'=-\sqrt{a^2-x^2}$$
So tangent is $$(y-k)=-\sqrt{a^2-h^2}(x-h)$$
Point where it cuts y-axis is $(0,k+h\sqrt{a^2-h^2})$
So the length is:
$$s=\sqrt{h^2+h^2(a^2-h^2)}=h\sqrt{1+h^2-a^2}$$ which isn't constant. I did everything correctly, what should be the correct way.
 [/INST] Let the point of tangency be $P(x_1,y_1)$. It is easy to check that the slope of the tangent line is
$$ m=y'\big|_{x=x_1}=-\frac{\sqrt{a^2-x_1^2}}{x_1}. $$
Hence the equation of the tangent line is
$$ y-y_1=--\frac{\sqrt{a^2-x_1^2}}{x_1}(x-x_1) $$
which cuts y-axis at $Q(0,y_1+\sqrt{a^2-x_1^2})$. Now
$$ |PQ|=\sqrt{(x_1-0)^2+(y_1-(y_1+\sqrt{a^2-x_1^2}))^2}=a.$$
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