Circle Geometry, SMB-001

In the circle centred at \(O\) the chord \(AB\) has length 7. The point \(E\) lies on \(AB\) and \(AE\) has length 4. The chord \(CD\) passes through \(E\).

Let the length of \(CD\) be \(\ell\) and the length of \(DE\) be \(x\).

Show that \(x^2-\ell x + 12 = 0\). (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

\(\text{See Worked Solutions}\)

Show Worked Solution

\(\text{Show}\ \ x^2-\ell x + 12 = 0\)

\(AB = 7 \ \ \Rightarrow \ EB=7-4=3\)

\(AE \times EB = ED \times CE\ \ \text{(intercepts of intersecting chords)}\)

\(4 \times 3\)	\(= x(\ell-x)\)
\(12\)	\(= x\ell-x^2\)
\(\therefore x^2-\ell x+12\)	\(=0\)