### Equations of circle and straight line, mid-point of intersections

Question Sample Titled 'Equations of circle and straight line, mid-point of intersections'

The coordinates of the centre of the circle ${X}$ are ${\left({9},{3}\right)}$ . It is given that the ${x}$-axis is a tangent to ${X}$ .

 (a) Find the equation of ${X}$ . (2 marks) (b) The slope and the ${y}$-intercept of the straight line ${L}$ is $-{4}$ and ${h}$ respectively. If ${L}$ cuts ${X}$ at ${T}$ and ${G}$ , express the coordinates of the mid-point of ${T}{G}$ in terms of ${h}$ .  (5 marks)

 (a) Note that the radius of ${X}$ is ${3}$ . 1M can be absorbed Thus, the equation of ${X}$ is ${\left({x}-{9}\right)}^{{2}}+{\left({y}-{3}\right)}^{{2}}={3}^{{2}}$ . 1A or equivalent (b) The equation of ${L}$ is ${y}=-{4}{x}+{h}$ . 1M Putting the equation of ${L}$ into the equation of ${X}$, which can be rewritten as ${x}^{{2}}+{y}^{{2}}-{18}{x}-{6}{y}+{81}={0}$ , we have ${x}^{{2}}+{\left(-{4}{x}+{h}\right)}^{{2}}-{18}{x}-{6}{\left(-{4}{x}+{h}\right)}+{81}$ $={0}$ . 1M So, we have ${17}{x}^{{2}}+{\left({6}-{8}{h}\right)}{x}+{\left({h}^{{2}}-{6}{h}+{81}\right)}$ $={0}$ . The ${x}$-coordinate of the mid-point of ${T}{G}$ $=\dfrac{{\dfrac{{-{\left({6}-{8}{h}\right)}}}{{17}}}}{{2}}$ sum of roots 1M $=\dfrac{{-{3}+{4}{h}}}{{17}}$ 1A The ${y}$-coordinate of the mid-point of ${T}{G}$ $=-{4}{\left(\dfrac{{-{3}+{4}{h}}}{{17}}\right)}+{h}$ $=\dfrac{{{12}+{h}}}{{17}}$ Thus, the required coordinates are ${\left(\dfrac{{-{3}+{4}{h}}}{{17}},\dfrac{{{12}+{h}}}{{17}}\right)}$ . 1A f.t.

# 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