### Mid point of intersections of circle and line

Question Sample Titled 'Mid point of intersections of circle and line'

The coordinates of the centre of a circle ${C}$ are ${\left({4},{6}\right)}$. It is given that the horizontal line ${y}={1}$ is a tangent of ${C}$.

 (a) Find the equation of ${C}$.  (2 marks) (b) ${L}$ is a straight line with slope $-\dfrac{{1}}{{2}}$ and ${x}$-intercept ${k}$ . ${L}$ cuts ${C}$ at two different points ${A}$ and ${B}$. Express the coordinates of the mid-point of ${A}$ and ${B}$ in terms of ${k}$.  (5 marks)

 (a) ∵  ${y}={1}$ is a tangent of ${C}$. ∴   The radius of circle$={6}-{1}={5}$ units 1A Equation of ${C}$ is: ${\left({x}-{4}\right)}^{{2}}+{\left({y}-{6}\right)}$ $={5}^{{2}}$ ${\left({x}-{4}\right)}^{{2}}+{\left({y}-{6}\right)}$ $={25}$ 1A  (b) Equation of ${L}$: $\dfrac{{{y}-{0}}}{{{x}-{k}}}$ $=-\dfrac{{1}}{{2}}$ 1M ${x}+{2}{y}-{k}$ $={0}$ Let ${M}{\left({a},{b}\right)}$ be the mid-pt. of ${A}$ and ${B}$, and ${O}$ be the centre of ${C}$. ${O}{M}\bot{A}{B}$ line joining centre to mid-pt. of chord ⊥ chord 1M Slope of ${O}{M}$ $=-{1}\div{\left(-\dfrac{{1}}{{2}}\right)}$  $={2}$ ∴  $\dfrac{{{b}-{6}}}{{{a}-{4}}}$ $={2}$ ${b}-{6}$ $={2}{a}-{8}$ ${2}{a}-{b}-{2}$ $={0}$ $\ldots{\left({1}\right)}$ 1M ∵  ${M}$ is a point on ${A}{B}$. ∴  ${a}+{2}{b}-{k}={0}$ $\ldots{\left({2}\right)}$ Solving, we have ${a}$ $=\dfrac{{{k}+{4}}}{{5}}$ 1A ${b}$ $=\dfrac{{{2}{k}-{2}}}{{5}}$ ∴   Required coordinates $={\left(\dfrac{{{k}+{4}}}{{5}},\dfrac{{{2}{k}-{2}}}{{5}}\right)}$ 1A

# 專業備試計劃

Level 4+ 保證及 5** 獎賞

ePractice 會以電郵、Whatsapp 及電話提醒練習

ePractice 會定期提供溫習建議

Level 5** 獎勵：會員如在 DSE 取得數學 Level 5** ，將獲贈一套飛往英國、美國或者加拿大的來回機票，唯會員須在最少 180 日內每天在平台上答對 3 題 MCQ。

Level 4 以下賠償：會員如在 DSE 未能達到數學 Level 4 ，我們將會全額退回所有會費，唯會員須在最少 180 日內每天在平台上答對 3 題 MCQ。

# FAQ

ePractice 是甚麼？

ePractice 是一個專為中四至中六而設的網站應用程式，旨為協助學生高效地預備 DSE 數學（必修部分）考試。由於 ePractice 是網站應用程式，因此無論使用任何裝置、平台，都可以在瀏覽器開啟使用。更多詳情請到簡介頁面。

ePractice 可以取代傳統補習嗎？

1. 會員服務期少於兩個月；或
2. 交易額少於 HK\$100。

Initiating...

HKDSE 數學試題練習平台