Consider the circle C : x^2+y^2-6 x-8 y-11=0

Question

Consider the circle C : $x^2+y^2-6 x-8 y-11=0$

Accepted Answer

Assumptions :

$F(h,k)$ is the foot of the \perp  drop from origin on the variable chord $AB$

The end-points of the chord $AB$ are

$\begin{gathered} A B \ ({x}_{1},{y}_{1}) ({x}_{2},{y}_{2}) \end{gathered}$

$\begin{gathered} \quad  \quad  \quad  \ C \quad  {x}^{2}+{y}^{2}-6x-8y-11=0 \quad  (x-3{)}^{2}+(y-4{)}^{2}=36 \quad  (3,4) \quad  6 \end{gathered}$

We will now form the line equations of the \perp  drop $OF$ and the variable chord $AB$ :

${l}_{OF}:y=(\frac{k}{h})x\Rightarrow {l}_{AB}:y=-(\frac{h}{k})x+c....(i)$

The foot of the \perp  on line ${l}_{AB}$ defined in equation (i) is $F(h,k)$ :

$k=-(\frac{h}{k})\cdot (h)+c\Rightarrow c=\frac{{h}^{2}+{k}^{2}}{k}\Rightarrow {l}_{AB}:hx+ky={h}^{2}+{k}^{2}\ldots \ldots (ii)$

Equation (ii) which gives the equation of the variable chord $AB$ cuts the circle at $A({x}_{1},{y}_{1})$ and $B({x}_{2},{y}_{2})$ is substituted in the equation of the circle $C$ from chart 1 to form 2 quadratics in $x$ and $y$ respectively :

Quadratic in $x$ :

${x}^{2}+{y}^{2}-6x-8y-11=0\Rightarrow {k}^{2}{x}^{2}+{k}^{2}{y}^{2}-6{k}^{2}x-8{k}^{2}y-11{k}^{2}=0$

${k}^{2}{x}^{2}+{({h}^{2}+{k}^{2}-hx)}^{2}-6{k}^{2}x-8k({h}^{2}+{k}^{2}-hx)-11{k}^{2}=0$

$({h}^{2}+{k}^{2}){x}^{2}-2x{h({h}^{2}+{k}^{2})+3{k}^{2}-4hk}+{({h}^{2}+{k}^{2})}^{2}-8k({h}^{2}+{k}^{2})-11{k}^{2}=0\leftarrow $ Quadratic in $x$

${x}_{1}{x}_{2}=\frac{{({h}^{2}+{k}^{2})}^{2}-8k({h}^{2}+{k}^{2})-11{k}^{2}}{({h}^{2}+{k}^{2})}\ldots$. (iii) where ${x}_{1}$ and ${x}_{2}$ are the roots of the quadratic

Quadratic in $y$ :

${x}^{2}+{y}^{2}-6x-8y-11=0\Rightarrow {h}^{2}{x}^{2}+{h}^{2}{y}^{2}-6{h}^{2}x-8{h}^{2}y-11{h}^{2}=0$

${({h}^{2}+{k}^{2}-ky)}^{2}+{h}^{2}{y}^{2}-6h({h}^{2}+{k}^{2}-ky)-8{h}^{2}y-11{h}^{2}=0$

$({h}^{2}+{k}^{2}){y}^{2}-2y{k({h}^{2}+{k}^{2})+4{h}^{2}-3hk}+{({h}^{2}+{k}^{2})}^{2}-6h({h}^{2}+{k}^{2})-11{h}^{2}=0$ Quadratic in $y$

${y}_{1}{y}_{2}=\frac{{({h}^{2}+{k}^{2})}^{2}-6h({h}^{2}+{k}^{2})-11{h}^{2}}{{h}^{2}+{k}^{2}}\ldots .(iv)$ where ${y}_{1}$ and ${y}_{2}$ are the roots of the quadratic

The variable chord $AB$ with end-points $A({x}_{1},{y}_{1})$ and $B({x}_{2},{y}_{2})$

Solutions · 1