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<p>A circle $C$ is given by the equation $3x^2 + 3y^2 + 9x - 15y - 39 = 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">Find the coordinates of the centre of the circle and the radius of the circle.</span></div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Determine, with working, if the point $(-3, 5)$ lies inside or outside the circle.</span></div></li></ol>
<p>Another circle $M$, with centre $\left( \frac{\sqrt{123} - 3}{2}, 12 \right)$, touches the circle $C$ at one point. Find</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">the distance between the two centres.</span></div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-content"><span class="list-item-text">all possible equations of $M$ in the form $x^2 + y^2 + 2gx + 2fy + c = 0$.</span></div></li></ol>

A circle $C$ is given by the equation $3x^2 + 3y^2 + 9x - 15y - 39 = 0$.

Find the coordinates of the centre of the circle and the radius of the circle.

Determine, with working, if the point $(-3, 5)$ lies inside or outside the circle.

Another circle $M$, with centre $\left( \frac{\sqrt{123} - 3}{2}, 12 \right)$, touches the circle $C$ at one point. Find

all possible equations of $M$ in the form $x^2 + y^2 + 2gx + 2fy + c = 0$.

Finding the Centre and Radius of a Circle

Length of a Line Segment

Problems Involving Two Circles

Position of a Point Relative to a Circle

Coordinate Geometry

<p>A circle, ${{C}_{1}}$, passes through the points $A\left( 4,0 \right)$ and $B\left( 0,8 \right)$. Its centre, $P$, lies on the line $y=x+1$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that the centre of the circle, ${{C}_{1}}$ is $\left( 4,5 \right)$. Hence find the equation of the circle, ${{C}_{1}}$.</div></li></ol>
<p>A second circle, ${{C}_{2}}$, with the equation ${{x}^{2}}+{{y}^{2}}+ax+by+32=0$, has the same centre, $P$, as circle ${{C}_{1}}$.</p>
<ol class="abc-list" start="2"><li><div class="list-item-paragraph">Find the value of $a$ and of $b$.</div></li><li><div class="list-item-paragraph">Show that the circle ${{C}_{2}}$ lies within the circle ${{C}_{1}}$.</div></li></ol>

A circle, ${{C}_{1}}$, passes through the points $A\left( 4,0 \right)$ and $B\left( 0,8 \right)$. Its centre, $P$, lies on the line $y=x+1$.

A second circle, ${{C}_{2}}$, with the equation ${{x}^{2}}+{{y}^{2}}+ax+by+32=0$, has the same centre, $P$, as circle ${{C}_{1}}$.

Finding the Equation of a Circle

Perpendicular Bisectors

<p>A circle $L$ is given by the equation $4{{x}^{2}}+4{{y}^{2}}+8x-16y-29=0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1760186652492-474g15szp" data-list-item-path="item-1760186652492-474g15szp"><div class="list-item-paragraph">Find the coordinates of the centre of the circle and the radius of the circle.</div></li><li data-list-item-id="item-1760186653318-05rb8lxds" data-list-item-path="item-1760186653318-05rb8lxds"><div class="list-item-paragraph">Determine, with working, if the point $\left( -2,5 \right)$ lies inside or outside the circle.</div></li></ol>
<p>Another circle $M$, with centre $\left( 4,14 \right)$, touches the circle $L$ at one point. Find</p>
<ol class="abc-list" start="3"><li data-list-item-id="item-1760186671281-pfwkqh6uz" data-list-item-path="item-1760186671281-pfwkqh6uz"><div class="list-item-paragraph">the distance between the two centres.</div></li><li data-list-item-id="item-1760186671880-ivnzbsj40" data-list-item-path="item-1760186671880-ivnzbsj40"><div class="list-item-paragraph">all the possible equations of circle $M$ in the form of ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$.</div></li></ol>

A circle $L$ is given by the equation $4{{x}^{2}}+4{{y}^{2}}+8x-16y-29=0$.

Another circle $M$, with centre $\left( 4,14 \right)$, touches the circle $L$ at one point. Find

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