A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

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>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

Perpendicular Bisectors

2025 MSHS PRELIMS P2 Q7

<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="246.3728" data-proportional style="width:400px;height:246.3728px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:400px;--graph-aspect:400 / 246.3728;"></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>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

Line–Circle Intersections

<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>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 $4y - 3x + 50 = 0$ 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 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 ${{x}^{2}}+{{y}^{2}}-10x+6y+30=0$.

Reflections and Translations of Circles

<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 $\left( 12,6 \right)$.

Midpoint of a Line Segment

<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 $P$.

<p>Given that the equation of the circle is ${{x}^{2}}+{{y}^{2}}-4x+2y+k=0$ where&nbsp;$k$ is a constant. Determine the value of $k$ for each of the following cases:</p>
<ol class="abc-list"><li data-list-item-id="item-1760186279657-m8qbvhdj5"><div class="list-item-paragraph">The area of the circle is $9\pi $ square units. Also, state the centre of this circle.</div></li><li data-list-item-id="item-1760186280188-8njtdmg34"><div class="list-item-paragraph">The line $y=x$ is a tangent to the circle.</div></li></ol>

Given that the equation of the circle is ${{x}^{2}}+{{y}^{2}}-4x+2y+k=0$ where $k$ is a constant. Determine the value of $k$ for each of the following cases:

Finding an unknown in the circle equation

<p>The equation of a circle, $C$ is given by ${{x}^{2}}+{{y}^{2}}-4x+2y+1=0$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the coordinates of the centre, and radius of $C$.</div></li><li><div class="list-item-paragraph">Find the equation of the tangent that touches $C$ at $\left( 4,-1 \right)$.</div></li></ol>

The equation of a circle, $C$ is given by ${{x}^{2}}+{{y}^{2}}-4x+2y+1=0$.

Math Question Bank