List of top Mathematics Questions on Circles

PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If $OP = 13$ cm, then find the length AB and PA.

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Let the circle $x^2+y^2=4$ intersect the $x$-axis at points $A(a,0)$ and $B(b,0)$. Let $P(2\cos\alpha,2\sin\alpha)$, $0<\alpha<\frac{\pi}{2}$, and $Q(2\cos\beta,2\sin\beta)$ be two points on the circle such that $(\alpha-\beta)=\frac{\pi}{2}$. Then the point of intersection of lines $AQ$ and $BP$ lies on:

Let PQ and MN be two straight lines touching the circle $x^2+y^2-4x-6y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle AOB = \pi/3$. Then the locus of the point of intersection of the lines PQ and MN is:

PQ is tangent to a circle with centre O. If $OQ = a$, $OP = a + 2$ and $PQ = 2b$, then relation between $a$ and $b$ is

In the given figure, O is the centre of the circle. PQ and PR are tangents. Show that the quadrilateral PQOR is cyclic.

Prove that the tangent at any point of a circle is perpendicular to the radius through the point of contact.

If a regular hexagon ABCDEF circumscribes a circle, then prove that AB + CD + EF = BC + DE + FA.

In the given figure, PA and PB are tangents to a circle centred at O. If $ \angle OAB = 15^\circ $, then $ \angle APB $ equals :

In the given figure, PA and PB are tangents to a circle centred at O. If $ \angle AOB = 130^\circ $, then $ \angle APB $ is equal to :

Prove that the lengths of tangents drawn from an external point to a circle are equal.

Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that $ \angle PTQ = 2 \angle OPQ $.

If $TP$ and $TQ$ are two tangents to a circle with centre $O$ from an external point $T$ so that $\angle POQ = 120^{\circ}$, then $\angle PTQ$ is equal to :

In the given figure, $PA$ is a tangent from an external point $P$ to a circle with centre $O$. If $\angle POB = 125^{\circ}$, then $\angle APO$ is equal to :

Two concentric circles are of radii 5 cm and 4 cm. Find the length of the chord of the larger circle which touches the smaller circle.

In the given figure, $\Delta ABC$ is a right triangle in which $\angle B = 90^\circ$, $AB = 4$ cm and $BC = 3$ cm. Find the radius of the circle inscribed in the triangle $ABC$.

In the given figure, if a circle touches the side $QR$ of $\Delta PQR$ at $S$ and extended sides $PQ$ and $PR$ at $M$ and $N$ respectively, then prove that : $PM = \frac{1}{2}(PQ + QR + PR)$.

The absolute difference between the squares of the radii of the two circles passing through the point $ (-9, 4) $ and touching the lines $ x + y = 3 $ and $ x - y = 3 $, is equal to:

Let $ ABC $ be the triangle such that the equations of lines $ AB $ and $ AC $ are: $ 3y - x = 2 \quad \text{and} \quad x + y = 2, $ respectively, and the points $ B $ and $ C $ lie on the x-axis. If $ P $ is the orthocentre of the triangle $ ABC $, then the area of the triangle $ PBC $ is equal to:

Let $ C_1 $ be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let $ C_2 $ be the circle with center $ (1, 3) $ that touches $ C_1 $ externally at the point $ (\alpha, \beta) $. If $ (\beta - \alpha)^2 = \frac{m}{n} $, and $ \gcd(m, n) = 1 $, then $ m + n $ is equal to:

If the equation of a circle is $ 4x^2 + 4y^2 - 12x + 8y = 0 $, what is the radius of the circle?

PA and PB are tangents drawn to a circle with centre O. If $\angle AOB = 120^\circ$ and OA = 10 cm, then

(i) Find $\angle OPA$.
(ii) Find the perimeter of $\triangle OAP$.
(iii) Find the length of chord AB.

The given figure shows a circle with centre O and radius 4 cm circumscribed by $\triangle ABC$. BC touches the circle at D such that BD = 6 cm, DC = 10 cm. Find the length of AE.

What is the area of a circle with diameter 10 cm?

In the following figure chord MN and chord RS intersect at point D. If RD = 15, DS = 4, MD = 8, find DN by completing the following activity:

Activity :
$\therefore$ MD $\times$ DN = $\boxed{\phantom{SD}}$ $\times$ DS $\dots$ (Theorem of internal division of chords)
$\therefore$ $\boxed{\phantom{8}}$ $\times$ DN = 15 $\times$ 4
$\therefore$ DN = $\frac{\boxed{\phantom{60}}}{8}$
$\therefore$ DN = $\boxed{\phantom{7.5}}$