Circle touching axes

12. \begin{figure}[h] \end{figure} Figure 3 shows a circle \(C\)
\(C\) touches the \(y\)-axis and has centre at the point ( \(a , 0\) ) where \(a\) is a positive constant.

1. Write down an equation for \(C\) in terms of \(a\) Given that the point \(P ( 4 , - 3 )\) lies on \(C\),
2. find the value of \(a\)

1. A circle has equation

$$x ^ { 2 } - 6 x + y ^ { 2 } + 8 y + k = 0$$ where \(k\) is a positive constant. Given that the \(x\)-axis is a tangent to this circle,

1. find the value of \(k\). The circle meets the coordinate axes at the points \(R , S\) and \(T\).
2. Find the exact area of the triangle \(R S T\).
   \includegraphics[max width=\textwidth, alt={}, center]{515f245f-9c5b-4263-ab2c-0a4f96f3bff0-21_2647_1840_118_111}

1. Express \(\sqrt { 22.5 }\) in the form \(k \sqrt { 10 }\).
2. A circle has the equation

$$x ^ { 2 } + y ^ { 2 } + 8 x - 4 y + k = 0$$ where \(k\) is a constant.

1. Find the coordinates of the centre of the circle. Given that the \(x\)-axis is a tangent to the circle,
2. find the value of \(k\).

5 A circle with centre \(C ( - 5,6 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{66813123-3876-4484-aad1-4bfc09bb1508-6_444_698_372_680}

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. 1. Verify that the point \(P ( - 2,2 )\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(P\).
   3. The mid-point of \(P C\) is \(M\). Determine whether the point \(P\) is closer to the point \(M\) or to the origin \(O\).

6 The circle with centre \(C ( 5,8 )\) touches the \(y\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{dbc25177-4a28-480f-93d5-41acb2a2d28c-5_485_631_370_715}

1. Express the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. 1. Verify that the point \(A ( 2,12 )\) lies on the circle.
   2. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(s x + t y + u = 0\), where \(s , t\) and \(u\) are integers.
3. The points \(P\) and \(Q\) lie on the circle, and the mid-point of \(P Q\) is \(M ( 7,12 )\).
   1. Show that the length of \(C M\) is \(n \sqrt { 5 }\), where \(n\) is an integer.
   2. Hence find the area of triangle \(P C Q\).

2. \begin{figure}[h] \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\).
2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
3. Find, to 3 significant figures, the length of \(P T\).

1. The point \(A\) has coordinates \(( 2,5 )\) and the point \(B\) has coordinates \(( - 2,8 )\). Find, in cartesian form, an equation of the circle with diameter \(A B\).

\begin{figure}[h] \end{figure} The circle \(C\), with centre \(( a , b )\) and radius 5 , touches the \(x\)-axis at \(( 4,0 )\), as shown in Fig. 1.

1. Write down the value of \(a\) and the value of \(b\).
2. Find a cartesian equation of \(C\). A tangent to the circle, drawn from the point \(P ( 8,17 )\), touches the circle at \(T\).
3. Find, to 3 significant figures, the length of \(P T\).

7 The circle \(S\) has centre \(C ( 8,13 )\) and touches the \(x\)-axis, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{fddf5016-a5bd-42db-b5c4-f4980b8d9d67-4_444_755_356_641}

1. Write down an equation for \(S\), giving your answer in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. The point \(P\) with coordinates \(( 3,1 )\) lies on the circle.
   1. Find the gradient of the straight line passing through \(P\) and \(C\).
   2. Hence find an equation of the tangent to the circle \(S\) at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
   3. The point \(Q\) also lies on the circle \(S\), and the length of \(P Q\) is 10 . Calculate the shortest distance from \(C\) to the chord \(P Q\).

3. $$\mathrm { f } ( x ) = x ^ { 3 } - ( k + 4 ) x + 2 k , \quad \text { where } k \text { is a constant. }$$ （a）Show that，for all values of \(k\) ，the curve with equation \(y = \mathrm { f } ( x )\) passes through the point \(( 2,0 )\) ．
（b）Find the values of \(k\) for which the equation \(\mathrm { f } ( x ) = 0\) has exactly two distinct roots． Given that \(k > 0\) ，that the \(x\)－axis is a tangent to the curve with equation \(y = \mathrm { f } ( x )\) ，and that the line \(y = p\) intersects the curve in three distinct points，
（c）find the set of values that \(p\) can take．
\includegraphics[max width=\textwidth, alt={}, center]{a243ceda-8175-4ae0-9bc7-b3048f468d10-3_573_899_343_704} The circle, with centre \(C\) and radius \(r\), touches the \(y\)-axis at \(( 0,4 )\) and also touches the line with equation \(4 y - 3 x = 0\), as shown in Fig. 1.

1. 1. Find the value of \(r\).
   2. Show that \(\arctan \left( \frac { 3 } { 4 } \right) + 2 \arctan \left( \frac { 1 } { 2 } \right) = \frac { 1 } { 2 } \pi\).
      (8) The line with equation \(4 x + 3 y = q , q > 12\), is a tangent to the circle.
2. Find the value of \(q\).
   (4)

6 A circle has centre \(C ( 10,4 )\). The \(x\)-axis is a tangent to the circle, as shown in Fig. 6. \begin{figure}[h] \end{figure}

1. Find the equation of the circle.
2. Show that the line \(y = x\) is not a tangent to the circle.
3. Write down the position vector of the midpoint of OC.