Let $C$ be a circle with a radius of 4, centered at the point $( 5,0 )$ on the $x$-axis. (1) If $\mathrm { P } ( p , q )$ is a point on circle $C$, then $$p ^ { 2 } - \mathbf { PQ } p + q ^ { 2 } + \mathbf { R } = 0 .$$ Also, the equation of the tangent to circle $C$ at point $\mathrm { P } ( p , q )$ is $$( p - \mathbf { S } ) x + q y = \mathbf { T } p - \mathbf { U } .$$ (2) Let us draw a line tangent to circle $C$ from point $\mathrm { A } ( 0 , a )$ on the $y$-axis, where $a \geqq 0$, and let $\mathrm { P } ( p , q )$ be the tangent point. The length of the segment AP is minimized at $a = \mathbf { V }$, and the length in this case is $\mathbf { W }$. Furthermore, the two tangents to circle $C$ from point A are orthogonal when the length of AP is $\mathbf { X }$. In this case, the value of $a$ is $a = \sqrt { \mathbf { Y } }$.
Let $C$ be a circle with a radius of 4, centered at the point $( 5,0 )$ on the $x$-axis.
(1) If $\mathrm { P } ( p , q )$ is a point on circle $C$, then
$$p ^ { 2 } - \mathbf { PQ } p + q ^ { 2 } + \mathbf { R } = 0 .$$
Also, the equation of the tangent to circle $C$ at point $\mathrm { P } ( p , q )$ is
$$( p - \mathbf { S } ) x + q y = \mathbf { T } p - \mathbf { U } .$$
(2) Let us draw a line tangent to circle $C$ from point $\mathrm { A } ( 0 , a )$ on the $y$-axis, where $a \geqq 0$, and let $\mathrm { P } ( p , q )$ be the tangent point.
The length of the segment AP is minimized at $a = \mathbf { V }$, and the length in this case is $\mathbf { W }$.
Furthermore, the two tangents to circle $C$ from point A are orthogonal when the length of AP is $\mathbf { X }$. In this case, the value of $a$ is $a = \sqrt { \mathbf { Y } }$.