Hard A-Level Circles Questions

Understand what the question is asking

Before diving in, we read the question carefully and picture the circle. Being clear on what we must find, and what we are given, stops us wasting effort.

Set up the right-angled triangle OAP

The tangent $AP$ meets the radius $OA$ at $90^{\circ}$, so triangle $OAP$ is right-angled at $A$.

State the radius

The circle has $r^2=20$, so $OA=\sqrt{20}=2\sqrt5$.

Find OP

The distance from the centre $O(0,0)$ to $P(0,10)$ is simply $10$.

Find the tangent length AP (Pythagoras)

Using the right angle at $A$, $AP^2=OP^2-OA^2$.

Simplify AP

So $AP=\sqrt{80}=4\sqrt5$.

Area of one triangle OAP

Triangle $OAP$ is right-angled at $A$, so its area is half the product of the two perpendicular sides $OA$ and $AP$.

Compute the triangle area

Since $2\sqrt5\times4\sqrt5=8\times5=40$, half of that is $20$.

The kite is two such triangles

By symmetry, triangle $OBP$ is identical, and together they make the kite $OAPB$.

State the area

So the area of the kite $OAPB$ is $40$ square units.

Check the answer looks sensible

It is always worth pausing to ask whether the answer is reasonable: a radius or length must be positive, and any coordinates should sit where we expect them on a sketch.

Re-read the question

A common way to lose marks is to answer a slightly different question. We re-read it and confirm we have given exactly what was requested, in the required form.

Recall the method used

The main tool here was tangent length. Recognising which circle property a question needs is the fastest way to know how to start next time.

Link to earlier topics

Circle work leans heavily on earlier skills such as completing the square, the distance formula and perpendicular gradients. Keeping those sharp makes these questions much easier.

Watch out for sign slips

Most mistakes in circle questions come from signs: $(y+3)$ means $b=-3$, and the right-hand side is $r^2$, not $r$. A quick sign check avoids these traps.

Understand what the question is asking

Before diving in, we read the question carefully and picture the circle. Being clear on what we must find, and what we are given, stops us wasting effort.

Complete the square on x

Halve $-2$ to get $-1$, write $(x-1)^2$ and subtract $1$.

Complete the square on y

Halve $4$ to get $2$, write $(y+2)^2$ and subtract $4$.

Rewrite the equation

Substitute the completed squares back into the equation.

Isolate the squared terms

Moving the constants across gives the standard form; here $r^2=5-k$.

The radius squared must be positive

For a genuine circle the radius must be a positive real number, so $r^2=5-k$ has to be greater than $0$.

Solve the inequality

Rearranging $5-k>0$ gives $k<5$.

Check the answer looks sensible

It is always worth pausing to ask whether the answer is reasonable: a radius or length must be positive, and any coordinates should sit where we expect them on a sketch.

Re-read the question

A common way to lose marks is to answer a slightly different question. We re-read it and confirm we have given exactly what was requested, in the required form.

Recall the method used

The main tool here was general form. Recognising which circle property a question needs is the fastest way to know how to start next time.

Link to earlier topics

Circle work leans heavily on earlier skills such as completing the square, the distance formula and perpendicular gradients. Keeping those sharp makes these questions much easier.

Watch out for sign slips

Most mistakes in circle questions come from signs: $(y+3)$ means $b=-3$, and the right-hand side is $r^2$, not $r$. A quick sign check avoids these traps.

Interpret the answer in words

Putting the result into plain words helps it stick: we can picture exactly what the number or equation tells us about the circle.

State the final answer clearly

Writing the final answer out on its own line makes sure the examiner can see it and that it is in the form the question asked for.

Sanity-check against a sketch

Comparing the answer with a quick sketch of the circle is a great final check: if the picture and the numbers disagree, we know to look again.

Understand what the question is asking

Before diving in, we read the question carefully and picture the circle. Being clear on what we must find, and what we are given, stops us wasting effort.

Substitute the line into the circle

Replace $y$ with $x+c$. Two distinct intersection points means the resulting quadratic has two real roots.

Expand

Expanding $(x+c)^2$ gives $x^2+2cx+c^2$.

Form a quadratic in x

Collect terms and move $8$ across; $c$ is treated as a constant.

Two distinct roots need a positive discriminant

For two different intersection points the discriminant must be strictly greater than zero.

Write the discriminant

Here $a=2$, $b=2c$ and $c_{\text{const}}=c^2-8$.

Expand

Multiply out: $4\times2\times(c^2-8)=8c^2-64$.

Simplify

Collecting the $c^2$ terms gives $-4c^2+64>0$.

Solve the inequality

Dividing by $-4$ flips the inequality sign, giving $c^2<16$.

Write the range

Taking square roots, $|c|<4$, so $-4<c<4$.

Check the answer looks sensible

It is always worth pausing to ask whether the answer is reasonable: a radius or length must be positive, and any coordinates should sit where we expect them on a sketch.

Re-read the question

A common way to lose marks is to answer a slightly different question. We re-read it and confirm we have given exactly what was requested, in the required form.

Recall the method used

The main tool here was line-circle intersection. Recognising which circle property a question needs is the fastest way to know how to start next time.

Link to earlier topics

Circle work leans heavily on earlier skills such as completing the square, the distance formula and perpendicular gradients. Keeping those sharp makes these questions much easier.

Watch out for sign slips

Most mistakes in circle questions come from signs: $(y+3)$ means $b=-3$, and the right-hand side is $r^2$, not $r$. A quick sign check avoids these traps.

Understand what the question is asking

Before diving in, we read the question carefully and picture the circle. Being clear on what we must find, and what we are given, stops us wasting effort.

Find where the circle meets the x-axis (y = 0)

Set $y=0$ in the equation. The $y$-terms vanish, leaving $x^2-6x=0$.

Solve for x

Factorising gives $x=0$ (the origin) and $x=6$, so $A=(6,0)$.

Find where the circle meets the y-axis (x = 0)

Set $x=0$. The $x$-terms vanish, leaving $y^2-8y=0$.

Solve for y

Factorising gives $y=0$ (the origin) and $y=8$, so $B=(0,8)$.

Recognise the right angle at O

$OA$ lies along the $x$-axis and $OB$ along the $y$-axis, so they meet at $90^{\circ}$ at the origin. (In fact $AB$ is a diameter — the angle in a semicircle!)

Use the right-angled triangle area

For a right-angled triangle the area is half the product of the two perpendicular sides.

Substitute the lengths

Here $OA=6$ and $OB=8$.

Compute

So the area of triangle $OAB$ is $24$ square units.

Check the answer looks sensible

It is always worth pausing to ask whether the answer is reasonable: a radius or length must be positive, and any coordinates should sit where we expect them on a sketch.

Re-read the question

A common way to lose marks is to answer a slightly different question. We re-read it and confirm we have given exactly what was requested, in the required form.

Recall the method used

The main tool here was intercepts of a circle. Recognising which circle property a question needs is the fastest way to know how to start next time.

Link to earlier topics

Circle work leans heavily on earlier skills such as completing the square, the distance formula and perpendicular gradients. Keeping those sharp makes these questions much easier.

Watch out for sign slips

Most mistakes in circle questions come from signs: $(y+3)$ means $b=-3$, and the right-hand side is $r^2$, not $r$. A quick sign check avoids these traps.

Interpret the answer in words

Putting the result into plain words helps it stick: we can picture exactly what the number or equation tells us about the circle.

Understand what the question is asking

Before diving in, we read the question carefully and picture the circle. Being clear on what we must find, and what we are given, stops us wasting effort.

Group the terms

Gather $x$ and $y$ terms and move the constant right, ready to complete the square and find the centre.

Complete the square on x

Halve $2$ to get $1$, write $(x+1)^2$ and subtract $1$.

Complete the square on y

Halve $-4$ to get $-2$, write $(y-2)^2$ and subtract $4$.

Form the standard equation

Combining constants: $20+1+4=25$, so $r^2=25$ and the centre is $(-1,2)$.

Confirm P lies on C

Check that $P(3,5)$ satisfies the equation before finding the tangent.

Gradient of the radius CP

The radius runs from the centre $(-1,2)$ to $P(3,5)$; its gradient is $\tfrac{3}{4}$.

Gradient of the tangent

The tangent is perpendicular to the radius, so we take the negative reciprocal of $\tfrac{3}{4}$.

Use y - y1 = m(x - x1)

Put the tangent gradient and $P(3,5)$ into the straight-line equation.

Expand

Multiplying out $-\tfrac{4}{3}(x-3)$ gives $-\tfrac{4}{3}x+4$.

Make y the subject

Adding $5$ to $4$ gives $9$, so the tangent is $y=-\tfrac{4}{3}x+9$.

Check the answer looks sensible

It is always worth pausing to ask whether the answer is reasonable: a radius or length must be positive, and any coordinates should sit where we expect them on a sketch.

Re-read the question

A common way to lose marks is to answer a slightly different question. We re-read it and confirm we have given exactly what was requested, in the required form.

Recall the method used

The main tool here was completing the square. Recognising which circle property a question needs is the fastest way to know how to start next time.

Link to earlier topics

Circle work leans heavily on earlier skills such as completing the square, the distance formula and perpendicular gradients. Keeping those sharp makes these questions much easier.