Let y=x be the equation of a chord of the circle C_1 (in the closed half-plane x 0 ) of di…

Question

Let $y=x$ be the equation of a chord of the circle ${C}_{1}$ (in the closed half-plane $x\ge 0$ ) of diameter 10 passing through the origin. Let ${C}_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle ${C}_{2}$, which passes through the point $(2,3)$ and is farthest from the center of ${C}_{2}$, is $x+ay+b=0$, then $a-b$ is equal to

Accepted Answer

To find the value of $a-b$,

Determine the equation of circle ${C}_{1}$

The circle ${C}_{1}$ has a diameter of 10 , so its radius is $R=5$. It passes through the origin ( 0,0 ) and is contained in the closed half-plane $x\ge 0$.

Let the center of ${C}_{1}$ be $(h,k)$. Since it passes through $(0,0)$, we have ${h}^{2}+{k}^{2}={R}^{2}=25$.

For the circle to lie entirely in the region $x\ge 0$ while passing through $(0,0)$, the distance from the center to the $y$-axis must be equal to the radius, and the center must have a non-negative $x$-coordinate.

Thus, $h=R=5$.

Substituting $h=5$ into ${h}^{2}+{k}^{2}=25$ gives $25+{k}^{2}=25$, which implies $k=0$.

The equation of ${C}_{1}$ is:

$(x-5{)}^{2}+{y}^{2}=25$

Find the center of circle ${C}_{2}$

The line $y=x$ is a chord of ${C}_{1}$. We find the intersection points of $y=x$ and ${C}_{1}$ :

$(x-5{)}^{2}+{x}^{2}=25$

$\Rightarrow$ ${x}^{2}-10x+25+{x}^{2}=25$

$\Rightarrow$$2{x}^{2}-10x=0$

$\Rightarrow$ $2x(x-5)=0$

The intersection points are $(0,0)$ and $(5,5)$. These are the endpoints of the diameter of circle ${C}_{2}$.

The center $M$ of ${C}_{2}$ is the midpoint of the chord:

$M=(\frac{0+5}{2},\frac{0+5}{2})=(2.5,2.5)$

Determine the equation of the chord of ${C}_{2}$

We are looking for the equation of a chord of ${C}_{2}$ passing through $P(2,3)$ that is farthest from the center $M(2.5,2.5)$.

In any circle, the chord passing through a point $P$ that is farthest from the center is the one perpendicular to the radius (or segment) $MP$.

The slope of the segment $MP$ is:

${m}_{MP}=\frac{3-2.5}{2-2.5}=\frac{0.5}{-0.5}=-1$

The slope $m$ of the required chord is the negative reciprocal of ${m}_{MP}$ :

$m=-\frac{1}{-1}=1$

Using the point-slope form for the line through $P(2,3)$ with slope $m=1$ :

$y-3=1(x-2)$

$\Rightarrow$ $y-3=x-2$
$\Rightarrow$ $x-y+1=0$

Calculate $a-b$

The equation is given in the form $x+ay+b=0$. Comparing this to $x-y+1=0$, we identify:

$\begin{gathered} a=-1 \ b=1 \end{gathered}$

Finally, we calculate the required value:

$a-b=-1-1=-2$

Solutions · 1