Pulley, cord, and tangent applications

CAIE P1 2020 June Q5

6 marks Standard +0.3

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\includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-06_761_460_258_840} The diagram shows a cord going around a pulley and a pin. The pulley is modelled as a circle with centre $O$ and radius 5 cm . The thickness of the cord and the size of the pin $P$ can be neglected. The pin is situated 13 cm vertically below $O$. Points $A$ and $B$ are on the circumference of the circle such that $A P$ and $B P$ are tangents to the circle. The cord passes over the major arc $A B$ of the circle and under the pin such that the cord is taut. Calculate the length of the cord.

CAIE P1 2010 November Q9

8 marks Standard +0.3

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\includegraphics[max width=\textwidth, alt={}, center]{73c0c113-8f35-4e7f-ad5d-604602498b71-4_837_1020_255_559} The diagram shows two circles, $C _ { 1 }$ and $C _ { 2 }$, touching at the point $T$. Circle $C _ { 1 }$ has centre $P$ and radius 8 cm ; circle $C _ { 2 }$ has centre $Q$ and radius 2 cm . Points $R$ and $S$ lie on $C _ { 1 }$ and $C _ { 2 }$ respectively, and $R S$ is a tangent to both circles.

Show that $R S = 8 \mathrm {~cm}$.
Find angle $R P Q$ in radians correct to 4 significant figures.
Find the area of the shaded region.

Edexcel C12 2014 June Q11

15 marks Challenging +1.2

11. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-17_1000_956_264_500} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the circle $C$ with centre $Q$ and equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 2 y + 5 = 0$$

Find
1. the coordinates of $Q$,
2. the exact value of the radius of $C$. The tangents to $C$ from the point $T ( 8,4 )$ meet $C$ at the points $M$ and $N$, as shown in Figure 4.
Show that the obtuse angle $M Q N$ is 2.498 radians to 3 decimal places. The region $R$, shown shaded in Figure 4, is bounded by the tangent $T N$, the minor arc $N M$, and the tangent $M T$.
Find the area of region $R$.

Edexcel C2 2008 January Q8

11 marks Moderate -0.3

A circle $C$ has centre $M ( 6,4 )$ and radius 3 .
1. Write down the equation of the circle in the form
$$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$ \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-12_833_1276_605_322}
\end{figure} Figure 3 shows the circle $C$. The point $T$ lies on the circle and the tangent at $T$ passes through the point $P ( 12,6 )$. The line $M P$ cuts the circle at $Q$.
Show that the angle $T M Q$ is 1.0766 radians to 4 decimal places. The shaded region $T P Q$ is bounded by the straight lines $T P , Q P$ and the arc $T Q$, as shown in Figure 3.
Find the area of the shaded region $T P Q$. Give your answer to 3 decimal places. \section*{9.} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{13c5a854-baea-4875-82bc-86a19c3be09c-14_675_844_283_534}
\end{figure} Figure 4 shows an open-topped water tank, in the shape of a cuboid, which is made of sheet metal. The base of the tank is a rectangle $x$ metres by $y$ metres. The height of the tank is $x$ metres. The capacity of the tank is $100 \mathrm {~m} ^ { 3 }$.
Show that the area $A \mathrm {~m} ^ { 2 }$ of the sheet metal used to make the tank is given by $$A = \frac { 300 } { x } + 2 x ^ { 2 }$$
Use calculus to find the value of $x$ for which $A$ is stationary.
Prove that this value of $x$ gives a minimum value of $A$.
Calculate the minimum area of sheet metal needed to make the tank.