As shown in figure, AP is tangent to the circle at point A. Secant through P intersects chord AY in point X, such that $AP=PX=XY$. If $PQ=1$ and $QZ=8$, then find AX.

Hint: First of all, using tangent – secant theorem i.e. If a tangent and a secant are drawn to a circle from an exterior point, then the square of the measure of the is equal to the product of the measures of the secant and its external secant segment, we will find the value of AP. Then using the fact that that two chords intersect each other internally, so the product of their segments should be equal which can be given as, $AX\times XY=ZX\times \text{QX}$, we will find the missing values as per the expression and then from that we will find the value of AX.

Complete step-by-step answer:
In question we are given that AP is tangent to the circle at point A and a secant through point P intersects chord AY in point X, such that $AP=PX=XY$. Now, $PQ=1$ and $QZ=8$, we are asked to find the value of AX. The figure is given below,
 
Now, as the tangent and secant are drawn to a circle from an exterior point, we can use tangent – secant theorem which states that, If a tangent and a secant are drawn to a circle from an exterior point, then the square of the measure of tangent is equal to the product of the measures of the secant and its external secant segment. It can be seen mathematically as per our figure as,
$PQ\times PZ=P{{A}^{2}}$ ………………(i)
Where, AP is a tangent and PZ is a secant and they interest in point P.
Now, from figure we can see that $PZ=PQ+QZ$, so on substituting this value in expression (i) we will get,
$PQ\times \left( PQ+QZ \right)=P{{A}^{2}}$
Now, values of $PQ=1$ and $QZ=8$ , so on substituting these values we will get,
$1\left( 1+8 \right)=P{{A}^{2}}$
$\Rightarrow P{{A}^{2}}={{3}^{2}}$
On taking square root on both the sides we will get,
$\Rightarrow PA=3$ …………….(ii)
So, we can say that $AP=PX=XY=3$, from expression (ii)
Now, from figure we can see that chords AY and QZ intersect each other internally, so the product of their segments should be equal, which can be given mathematically as,
$\text{ }AY=QZ$
$\text{Segments of chord }AY=AX\times XY$
$\text{Segments of chord QZ}=ZX\times \text{QX}$
So, on substituting the values we will get,
$AX\times XY=ZX\times \text{QX}$ ………………….(iii)
Now, we have derived the value of XY, so to find the value of AX we have to find the value of ZX and QX. Now, from figure we can see that, $PX=PQ+QX$ , so on substituting values $PX=3$ and $PQ=1$, we will get,
$PX=PQ+QX$
$3=1+QX$
$\Rightarrow QX=3-1=2$
Same way, we can for ZX we can say that $QZ=ZX+QX$, where value of $QZ=8$ and $QX=2$ , so on substituting these values we will get,
$QZ=ZX+QX$
$\Rightarrow 8=ZX+2$
$\Rightarrow ZX=8-2=6$
Now, on substituting all the values of XY, ZX and QX in expression (iii) we will get,
$AX\times XY=ZX\times \text{QX}$
$\Rightarrow AX\times 3=6\times 2$
$\Rightarrow AX=\dfrac{6\times 2}{3}=2\times 2$
$\Rightarrow AX=4$
Thus, the value of AX is 4.

Note: Here, in tangent secant theorem we considered the formula, $PQ\times PZ=P{{A}^{2}}$, instead of that if we consider, $PQ\times PZ=PA$, then value of AP will become, $PA=1\left( 1+8 \right)=9$, so, due to that value of PX and XY also becomes 9 and ultimately our final answer also changes and becomes, $AX\times 8=0\times 8\Rightarrow AX=0$, as value of PX becomes 8 , so, value of ZX becomes 0 which can not be possible. So, students must use formula properly to get a proper solution.