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PSLE Foundation Mathematics

<p>Two points $ A(-3,-6) $ and $ B (7,-6) $ lie on a circle centre $ G$. The equation of the tangent to the circle at the point $ Q(1,-12) $ is $x+5y+59=0$.</p>
<ol class="abc-list"><li data-list-item-id="item-1773467964823-vim5kux87"><div class="list-item-paragraph">Show that the coordinates of $G$ is $ (2,-7) $<span class="inline-marks">[5]</span></div></li><li data-list-item-id="item-1773467967295-xi8wvogbp"><div class="list-item-paragraph">Find the equation of the circle in the form of $x^2+y^2+2gx+2fy+c=0$, stating the value of $f$, $g$ and  $c$.<span class="inline-marks">[3]</span></div></li></ol>

Two points $A(-3,-6)$ and $B(7,-6)$ lie on a circle with centre $G$. The equation of the tangent to the circle at the point $Q(1,-12)$ is $x+5y+59=0$.

Show that the coordinates of $G$ is $(2,-7)$.

Find the equation of the circle in the form of $x^2+y^2+2gx+2fy+c=0$, stating the value of $f$, $g$ and $c$.

Finding the Equation of a Circle

Tangents and Normals to Circles

Perpendicular Bisectors

2025 MSHS PRELIMS P2 Q7

Coordinate Geometry

<p>The line $y + 3x = 6$ intersects the circle, $C_{1}$, $x^{2} + y^{2} - 2x - 4y = 11$ at $A$ and $B$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">State the centre and find the radius of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of $A$ and $B$ and state if&nbsp;$AB$ is the diameter of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the circle $C_{2}$, when&nbsp;$C_{1}$ is reflected along the line $x = 3$.</span></div></li></ol>

The line $y + 3x = 6$ intersects the circle, $C_{1}$, $x^{2} + y^{2} - 2x - 4y = 11$ at $A$ and $B$.

State the centre and find the radius of the circle.

Find the coordinates of $A$ and $B$ and state if $AB$ is the diameter of the circle.

Find the equation of the circle $C_{2}$, when $C_{1}$ is reflected along the line $x = 3$.

Finding the Centre and Radius of a Circle

Reflections and Translations of Circles

Line–Circle Intersections

<p>A circle with centre $C$ passes through points $A(-8, 3)$ and $B(-1, -4)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the perpendicular bisector of $AB$ will pass through $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$C$ lies on the line $y = 2x + 5$, find $C$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the equation of the circle.</span></div></li></ol>

A circle with centre $C$ passes through points $A(-8, 3)$ and $B(-1, -4)$.

Explain why the perpendicular bisector of $AB$ will pass through $C$.

Given that $C$ lies on the line $y = 2x + 5$, find $C$.

Intersection of Two Lines

<p>The points $A(7, 7)$ and $B(-2, -5)$ are at the opposite ends of a diameter of a circle $C_1$. Find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">centre of the circle and the radius of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">equation of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">equation of the perpendicular bisector $AB$.</span></div></li></ol>
<p>The circle, $C_1$, is reflected in the $x$-axis.</p>
<ol class="abc-list" start="4"><li><div class="list-item-content"><span class="list-item-text">Find the equation of the reflected circle, $C_2$.</span></div></li></ol>

The points $A(7, 7)$ and $B(-2, -5)$ are at the opposite ends of a diameter of a circle $C_1$. Find the

centre of the circle and the radius of the circle.

equation of the perpendicular bisector $AB$.

The circle, $C_1$, is reflected in the $x$-axis.

Find the equation of the reflected circle, $C_2$.

<p>A circle $C$ passes through the point $P \ (-6, 6)$ and has the same centre as the circle $x^{2} + y^{2} + 6x - 4y + 2 = 0$.</p>
<ol class="abc-list"><li data-list-item-id="item-a"><div class="list-item-content"><span class="list-item-text">Find the equation of the circle $C$.</span></div></li><li data-list-item-id="item-b"><div class="list-item-paragraph">Find the equation of the tangent to the circle $C$, at the point $ (-6,6) $.</div></li></ol>
<p>Another point $Q \ (-6, q)$ which lies on circle $C$, is the same distance from the $y$-axis as the point $P$.</p>
<ol class="abc-list" start="4"><li data-list-item-id="item-c"><div class="list-item-content"><span class="list-item-text">Find the coordinates of the point $Q$.</span></div></li><li data-list-item-id="item-d"><div class="list-item-content"><span class="list-item-text">Find the perimeter of the triangle $POQ$, where&nbsp;$O$ is the centre of the circle $C$.</span></div></li></ol>

A circle $C$ passes through the point $P \ (-6, 6)$ and has the same centre as the circle $x^{2} + y^{2} + 6x - 4y + 2 = 0$.

Another point $Q \ (-6, q)$ which lies on circle $C$, is the same distance from the $y$-axis as the point $P$.

Find the perimeter of the triangle $POQ$, where $O$ is the centre of the circle $C$.

Length of a Line Segment

<p>Find the equation of the circle whose centre is $\left( 3,8 \right)$ and which touches the $y$-axis.</p>

Find the equation of the circle whose centre is $\left( 3,8 \right)$ and which touches the $y$-axis.

<p>Find the equation of the circle with diameter $AB$, where coordinates of $A$ and $B$ are $\left( 5,-9 \right)$ and $\left( 3,6 \right)$ respectively.</p>

Find the equation of the circle with diameter $AB$, where coordinates of $A$ and $B$ are $\left( 5,-9 \right)$ and $\left( 3,6 \right)$ respectively.

Midpoint of a Line Segment

<p>Find the equation of the circle with centre $C\left( 4,5 \right)$ and a point on the circle of $\left( -3,8 \right)$.</p>

Find the equation of the circle with centre $C\left( 4,5 \right)$ and a point on the circle of $\left( -3,8 \right)$.

<p>Find the equation of the circle with centre $C\left( 4,8 \right)$ and radius $7$.</p>

Find the equation of the circle with centre $C\left( 4,8 \right)$ and radius $7$.

<p>The line $4y - 3x + 50 = 0$ is a tangent to a circle at the point $A$. The centre of the circle is $C(3, -4)$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the coordinates of $A$ is $(6, -8)$.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the radius of the circle and the equation of the circle.</span><span class="inline-marks">[3]</span></div></li></ol>

The line $4y - 3x + 50 = 0$ is a tangent to a circle at the point $A$. The centre of the circle is $C(3, -4)$.

Show that the coordinates of $A$ is $(6, -8)$.

Hence find the radius of the circle and the equation of the circle.

2023 ACSBR AM S3 EOY Q5 

<p>A tangent to a circle at the point ${(3, 1)}$ intersects the x-axis at ${x = 5}$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the tangent ${T}$.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-paragraph">The centre of the circle lies on the line ${y = 15 - 3x}$. Find the equation of the circle.<span class="inline-marks">[6]</span></div></li><li><div class="list-item-paragraph">Find the equations of the tangents to the circle parallel to the y-axis.<span class="inline-marks">[3]</span></div></li></ol>

A tangent to a circle at the point ${(3, 1)}$ intersects the x-axis at ${x = 5}$.

Equation of a Straight Line

2023 AISS AM S3 EOY Q10

<p>The line $y+2x=5$ intersects the circle, ${{C}_{1}}$, ${{x}^{2}}+{{y}^{2}}-4x-6y=12$ at $A$ and $B$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">State the centre of the circle and find the radius of the circle.</div></li><li><div class="list-item-paragraph">Given that the coordinates of $A$ and $B$ are $\left( -1,7 \right)$ and $\left( \frac{17}{5},-\frac{9}{5} \right)$ respectively, show that&nbsp;$AB$ is not a diameter of the circle.</div></li><li><div class="list-item-paragraph">Find the equation of the circle, ${{C}_{2}}$, when&nbsp;${{C}_{1}}$ is reflected along the line $x=1$.</div></li></ol>

The line $y+2x=5$ intersects the circle, ${{C}_{1}}$, ${{x}^{2}}+{{y}^{2}}-4x-6y=12$ at $A$ and $B$.

<p>The points $A(7,-6)$, $B\left( 9,2 \right)$ and $C\left( 1,4 \right)$ lie on a circle as shown in the diagram.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Find the gradients of $AB$ and $BC$ and show that&nbsp;$AC$ is the diameter of the circle.</div></li><li><div class="list-item-paragraph">Find the coordinates of the centre of the circle and find the radius of the circle.</div></li><li><div class="list-item-paragraph">Find the equation of the circle.</div></li><li><div class="list-item-paragraph">Find the equation of the perpendicular bisector of $BC$ and determine whether it passes through the centre of the circle.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768288048546-x407cmgr62.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyODgwNDg1NDYteDQwN2NtZ3I2Mi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODUzLCJleHAiOjE3OTI2OTYxNTN9.wnoPWgISL1w537HKQf_2AqZyuW-lWIlxrmiOQpV3IMY" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

The points $A(7,-6)$, $B\left( 9,2 \right)$ and $C\left( 1,4 \right)$ lie on a circle as shown in the diagram.

Parallel and Perpendicular Lines

Gradient of a Straight Line

<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 ${{C}_{1}}$.

Problems Involving Two Circles

<p>A circle with centre $C$ passes through points $A\left( -1,7 \right)$ and $B\left( 0,8 \right)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Explain why the perpendicular bisector of $AB$ will pass through $C$.</div></li><li><div class="list-item-paragraph">Given that $C$ lies on the line $y=2x-2$, show that $C$ is $\left( 3,4 \right)$.</div></li><li><div class="list-item-paragraph">Hence find the equation of the circle.</div></li></ol>

A circle with centre $C$ passes through points $A\left( -1,7 \right)$ and $B\left( 0,8 \right)$.

<p>The equation of the circle ${{C}_{1}}$ is given by ${{x}^{2}}+{{y}^{2}}-10x+6y+30=0$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the coordinates of its center and the length of its radius.</div></li><li><div class="list-item-paragraph">Show that the circle ${{C}_{1}}$ touches the line $x=3$.</div></li><li><div class="list-item-paragraph">The circle, ${{C}_{1}}$ is reflected about the line$x=1$. Find the equation of the reflected circle, ${{C}_{2}}$.</div></li><li><div class="list-item-paragraph">Find the equation of another circle ${{C}_{3}}$ which has an area that is $4$ times larger than ${{C}_{1}}$ and its center at the origin.</div></li></ol>

The equation of the circle ${{C}_{1}}$ is given by ${{x}^{2}}+{{y}^{2}}-10x+6y+30=0$.

<p>The points $A\left( -2,4 \right)$ and $B\left( 6,-2 \right)$ are at the opposite ends of a diameter of a circle, ${{C}_{1}}$. Find the</p>
<ol class="abc-list"><li><div class="list-item-paragraph">centre of the circle and radius of the circle.</div></li><li><div class="list-item-paragraph">equation of the circle.</div></li><li><div class="list-item-paragraph">equation of the perpendicular bisector of $AB$.</div></li></ol>
<p>The circle, ${{C}_{1}}$, is reflected in the $x$-axis.</p>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Find the equation of the reflected circle, ${{C}_{2}}$.</div></li></ol>

The points $A\left( -2,4 \right)$ and $B\left( 6,-2 \right)$ are at the opposite ends of a diameter of a circle, ${{C}_{1}}$. Find the

The circle, ${{C}_{1}}$, is reflected in the $x$-axis.

<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

Position of a Point Relative to a Circle

<p>A circle, ${{C}_{1}}$, has a diameter $AB$ where $A$ is the point $(6,-2)$ and $B$ is the point $\left( 12,6 \right)$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the coordinates of the centre and the radius of ${{C}_{1}}$.</div></li><li><div class="list-item-paragraph">Find the equation of ${{C}_{1}}$.</div></li><li><div class="list-item-paragraph">Show that the equation of the tangent to the circle at $A$ is $4y+3x=10.$</div></li></ol>

A circle, ${{C}_{1}}$, has a diameter $AB$ where $A$ is the point $(6,-2)$ and $B$ is the point $\left( 12,6 \right)$.

<p>A circle $C$ passes through the point $P\left( 4,-3 \right)$ and has the same centre as the circle ${{x}^{2}}+{{y}^{2}}+4x-2y-1=0$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the equation of the circle $C$.</div></li><li><div class="list-item-paragraph">Find the equation of the tangent to the circle $C$ at the point $P\left( 4,-3 \right)$.</div></li></ol>
<p>Another point $Q\left( 4,q \right)$ which lies on the circle $C$, is the same distance from the $y$-axis as the point $P$.</p>
<ol class="abc-list" start="3"><li><div class="list-item-paragraph">Find the coordinates of the point $Q$ .</div></li><li><div class="list-item-paragraph">Find the perimeter of the triangle $POQ$, where $O$ is the centre of the circle $C$.</div></li></ol>

A circle $C$ passes through the point $P\left( 4,-3 \right)$ and has the same centre as the circle ${{x}^{2}}+{{y}^{2}}+4x-2y-1=0$.

Another point $Q\left( 4,q \right)$ which lies on the circle $C$, is the same distance from the $y$-axis as the point $P$.

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