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PSLE Foundation Mathematics

<p>A circle has a diameter $AB$. The point $A$ has coordinates $(1, -6)$ and the equation of the tangent to the circle at $B$ is $3x + 4y = k$.</p>
<ol class="abc-list"><li data-list-item-id="item-1" data-list-item-path="item-1"><div class="list-item-paragraph">Show that the equation of the normal to the circle at the point $A$ is $4x - 3y = 22$.<span class="inline-marks">[3]</span></div></li></ol>
<p>It is also given also that the line $x = -1$ touches the circle at the point $(-1, -2)$.</p>
<ol class="abc-list" start="2"><li data-list-item-id="item-2" data-list-item-path="item-2"><div class="list-item-paragraph">Find the coordinates of the centre and the radius of the circle.<span class="inline-marks">[4]</span></div></li><li data-list-item-id="item-3" data-list-item-path="item-3"><div class="list-item-paragraph">Find the value of $k$.<span class="inline-marks">[3]</span></div></li></ol>

A circle has a diameter $AB$. The point $A$ has coordinates $(1, -6)$ and the equation of the tangent to the circle at $B$ is $3x + 4y = k$.

It is also given also that the line $x = -1$ touches the circle at the point $(-1, -2)$.

Finding the Centre and Radius of a Circle

Tangents and Normals to Circles

2024 CGSS P2 Q8

Coordinate Geometry

<p>A pendulum bob and string with tangential direction is modelled by the equation $2y + 4x = -20$.</p>
<div class="graph-block" data-graph-config="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" data-width="400" data-height="200" data-proportional style="width:400px;height:200px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:0px;--graph-width:400px;--graph-aspect:400 / 200;"></div>
<p>A physicist spilled coffee on his notes and the equation for the curved path of the pendulum bob became $x^2 + y^2 + ax - 2y + 5 = 0$, where&nbsp;$a$ is now an unknown constant.</p>
<p>Find the possible lengths of the pendulum string in centimetres.</p>

A pendulum bob and string with tangential direction is modelled by the equation $2y + 4x = -20$.

A physicist spilled coffee on his notes and the equation for the curved path of the pendulum bob became $x^2 + y^2 + ax - 2y + 5 = 0$, where $a$ is now an unknown constant.

Find the possible lengths of the pendulum string in centimetres.

<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 Equation of a Circle

Reflections and Translations of Circles

Line–Circle Intersections

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the coordinates of the centre and the radius of the circle $x^{2} + y^{2} - 4x - 8y - 5 = 0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the circle cuts the $x$-axis at $P$ and $Q$, find the equation of the perpendicular bisector of the line $PQ$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the line $2y - 6x + 4 = 0$ intersects the circle at $R$ and $S$, find the coordinates of $R$ and $S$. Hence show that the line segment $RS$ is a diameter of the circle.</span></div></li></ol>

Find the coordinates of the centre and the radius of the circle $x^{2} + y^{2} - 4x - 8y - 5 = 0$.

If the circle cuts the $x$-axis at $P$ and $Q$, find the equation of the perpendicular bisector of the line $PQ$.

Given that the line $2y - 6x + 4 = 0$ intersects the circle at $R$ and $S$, find the coordinates of $R$ and $S$. Hence show that the line segment $RS$ is a diameter of the circle.

Midpoint of a Line Segment

Perpendicular Bisectors

<p>The point $A(2, 1)$ lies on the circle $x^2 + y^2 + kx + 4y + 7 = 0$, where&nbsp;$k$ is a constant.</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the value of $k$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the centre and radius of the circle,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the coordinates of $B$ where&nbsp;$AB$ is the diameter of the circle.</span></div></li></ol>

The point $A(2, 1)$ lies on the circle $x^2 + y^2 + kx + 4y + 7 = 0$, where $k$ is a constant.

the coordinates of $B$ where $AB$ is the diameter of the circle.

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the radius and coordinates of the centre of the circle $x^2 + y^2 - 4x - 6y - 12 = 0$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$AB$ is a diameter of this circle, where&nbsp;$A$ is the point $(2, 8)$, find the coordinates of the point $B$.</span></div></li></ol>

Find the radius and coordinates of the centre of the circle $x^2 + y^2 - 4x - 6y - 12 = 0$.

Given that $AB$ is a diameter of this circle, where $A$ is the point $(2, 8)$, find the coordinates of the point $B$.

<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 $C$ 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 $x^2 + y^2 + 2gx + 2fy + c = 0$.

Length of a Line Segment

Problems Involving Two Circles

Position of a Point Relative to a Circle

<p>A circle has equation $x^2 + y^2 + 22x + 40y - 8 = 0$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the centre and radius of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Is the point $(23, 15)$ inside the circle? Explain your answer clearly.</span></div></li></ol>

A circle has equation $x^2 + y^2 + 22x + 40y - 8 = 0$.

Find the centre and radius of the circle.

Is the point $(23, 15)$ inside the circle? Explain your answer clearly.

<p>Given that the equation of the circle is $x^{2} + y^{2} + 12x - 16y + k = 0$ where&nbsp;$k$ is a constant.<br>Determine the value of $k$ for each of the following cases:</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The area of the circle is $81\pi$ square units. Also, state the centre of the circle.</span></div></li><li><div class="list-item-content"><span class="list-item-text">The line $y = x$ is a tangent to the circle.</span></div></li></ol>

Given that the equation of the circle is $x^{2} + y^{2} + 12x - 16y + k = 0$ where $k$ is a constant.
Determine the value of $k$ for each of the following cases:

The area of the circle is $81\pi$ square units. Also, state the centre of the circle.

The line $y = x$ is a tangent to the circle.

<p>The equation of a circle, $C$ is given by $x^{2} + y^{2} - 6x + 2y + 1 = 0$.</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, and radius of $C$.</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">Find the equation of the tangent that touches $C$ at $(6, -1)$.</span></div></li></ol>

The equation of a circle, $C$ is given by $x^{2} + y^{2} - 6x + 2y + 1 = 0$.

Find the coordinates of the centre, and radius of $C$.

Find the equation of the tangent that touches $C$ at $(6, -1)$.

<p>Find the centre and radius of a circle that has the equation $\left( 2x+6 \right)\left( x+3 \right)+\left( 2y+2 \right)\left( y-7 \right)=0$.</p>

Find the centre and radius of a circle that has the equation $\left( 2x+6 \right)\left( x+3 \right)+\left( 2y+2 \right)\left( y-7 \right)=0$.

<p>Find the coordinates of the centre and radius of the circle $3{{x}^{2}}+3{{y}^{2}}=6y-4x+7$.</p>

Find the coordinates of the centre and radius of the circle $3{{x}^{2}}+3{{y}^{2}}=6y-4x+7$.

<p>Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}=4x+6$.</p>

Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}=4x+6$.

<p>Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}-3x-8y+10=0$.</p>

Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}-3x-8y+10=0$.

<p>Find the coordinates of the centre and radius of the circle ${{\left( x-3 \right)}^{2}}+{{\left( y+8 \right)}^{2}}=13$.</p>

Find the coordinates of the centre and radius of the circle ${{\left( x-3 \right)}^{2}}+{{\left( y+8 \right)}^{2}}=13$.

<p>Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}=64$.</p>

Find the coordinates of the centre and radius of the circle ${{x}^{2}}+{{y}^{2}}=64$.

<p>A circle has the equation $x^2 + y^2 - 8x - 4y = 5$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the radius of the circle and the coordinates of its centre.</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">The line $y = x - 1$ intersects the circle at two points $M$ and $N$.</div><div class="list-item-paragraph">Show that the coordinates of $M$ and $N$ are $(0, -1)$ and $(7, 6)$ respectively.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">Determine if $MN$ is the diameter of the circle.</div><div class="list-item-paragraph">Show your working clearly.<span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the tangent of the circle at point $N$.</span><span class="inline-marks">[3]</span></div></li></ol>

A circle has the equation $x^2 + y^2 - 8x - 4y = 5$.

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

Find the equation of the tangent of the circle at point $N$.

2023 CSS AM S3 EOY Q11 

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgyODgwNDg1NDYteDQwN2NtZ3I2Mi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkzNTg2Njk3LCJleHAiOjE3OTM1ODY5OTd9.Sl3UeVTHc29SQdPBtKi7xX-GuwqhAt_XDEHx5klfDno" 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.

Math Question Bank