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<p>The curve $(x-1)^2+(y-3)^2 =5$ intersects the line $y-3x=5$ at two points. Find the coordinates of these two points.<span class="inline-marks">[5]</span></p>

The curve $(x-1)^2+(y-3)^2=5$ intersects the line $y-3x=5$ at two points. Find the coordinates of these two points.

Line–Circle Intersections

2025 MSHS PRELIMS P1 Q1

Coordinate Geometry

<p>The line $y + x = 1$ intersects the curve $x^2 + y^2 - 2y = 7$ at $P$ and $Q$. Find the equation of the perpendicular bisector $PQ$.</p>

The line $y + x = 1$ intersects the curve $x^2 + y^2 - 2y = 7$ at $P$ and $Q$. Find the equation of the perpendicular bisector $PQ$.

Perpendicular Bisectors

<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

Finding the Equation of a Circle

Reflections and Translations of Circles

<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

<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

Tangents and Normals to Circles

<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 

<ol class="abc-list"><li><div class="list-item-paragraph">Find the coordinates of the centre and the radius of the circle ${{x}^{2}}+{{y}^{2}}-2x+4y-8=0$.</div></li><li><div class="list-item-paragraph">If the circle cuts the $x$-axis at $P$ and $Q$, find the equation of the perpendicular bisector of the line $PQ$.</div></li><li><div class="list-item-paragraph">Given that the line $2y-3x+7=0$ 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.</div></li></ol>

<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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