1.03d Circles: equation (x-a)^2+(y-b)^2=r^2

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).

2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\) WIHN SIHI NITIIUM ION OC
   VIUV SIHI NI JAHM ION OC
   VI4V SIHI NIS IIIM ION OC

A circle, with centre \(C\) and radius \(r\), has equation $$x ^ { 2 } + y ^ { 2 } - 8 x + 4 y - 12 = 0$$ Find

1. the coordinates of \(C\),
2. the exact value of \(r\). The circle cuts the \(y\)-axis at the points \(A\) and \(B\).
3. Find the coordinates of the points \(A\) and \(B\).

Solve the simultaneous equations $$\begin{gathered} y - 2 x - 4 = 0 \\ 4 x ^ { 2 } + y ^ { 2 } + 20 x = 0 \end{gathered}$$

The points \(A\) and \(B\) have coordinates \(( 5 , - 1 )\) and \(( 13,11 )\) respectively.

1. Find the coordinates of the mid-point of \(A B\). Given that \(A B\) is a diameter of the circle \(C\),
2. find an equation for \(C\).

The circle \(C\) has centre \(( 3,4 )\) and passes through the point \(( 8 , - 8 )\). Find an equation for C

3. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 8 \cos \theta , 6 \sin \theta )\).

1. Using calculus, show that an equation for \(l\) is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
   The point \(M\) is the midpoint of \(A B\).
2. Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.

7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).

1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\) The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\) Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)

1. The line \(x = 8\) is a directrix of the ellipse with equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , b > 0$$ and the point \(( 2,0 )\) is the corresponding focus.
Find the value of \(a\) and the value of \(b\).

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find

1. the coordinates of the foci of \(H\),
2. the equations of the directrices of \(H\).

1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).

Find a cartesian equation for \(H\).

1. The point \(P\) lies on the ellipse \(E\) with equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ \(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\) \(M\) is the midpoint of \(P N\).

1. Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
2. Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
3. Show that this locus is a circle and state its centre and radius.

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$ The foci of \(H\) are at the points with coordinates \(( 13,0 )\) and \(( - 13,0 )\).
Find

1. the value of the constant \(a\),
2. the equations of the directrices of \(H\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c532661c-8a94-483a-a921-b35d5c0a0188-04_754_810_1053_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.

1. Calculate the gradient of \(D E\).
2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
4. Calculate the length of \(D F\).
5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).

5

1. Express \(x ^ { 2 } + 3 x\) in the form \(( x + a ) ^ { 2 } + b\).
2. Express \(y ^ { 2 } - 4 y - \frac { 11 } { 4 }\) in the form \(( y + p ) ^ { 2 } + q\). A circle has equation \(x ^ { 2 } + y ^ { 2 } + 3 x - 4 y - \frac { 11 } { 4 } = 0\).
3. Write down the coordinates of the centre of the circle.
4. Find the radius of the circle.

10 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 2 x - 4 y - 8 = 0\).

1. Find the centre and radius of the circle.
2. The circle passes through the point \(( - 3 , k )\), where \(k < 0\). Find the value of \(k\).
3. Find the coordinates of the points where the circle meets the line with equation \(x + y = 6\).

2

1. Write down the equation of the circle with centre \(( 0,0 )\) and radius 7 .
2. A circle with centre \(( 3,5 )\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 10 y - 30 = 0\). Find the radius of the circle.

8

1. Describe completely the curve \(x ^ { 2 } + y ^ { 2 } = 25\).
2. Find the coordinates of the points of intersection of the curve \(x ^ { 2 } + y ^ { 2 } = 25\) and the line \(2 x + y - 5 = 0\).

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

9

1. Find the equation of the circle with radius 10 and centre ( 2,1 ), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. The circle passes through the point \(( 5 , k )\) where \(k > 0\). Find the value of \(k\) in the form \(p + \sqrt { q }\).
3. Determine, showing all working, whether the point \(( - 3,9 )\) lies inside or outside the circle.
4. Find an equation of the tangent to the circle at the point ( 8,9 ).

7 \includegraphics[max width=\textwidth, alt={}, center]{5fa27228-37b2-45d9-a8dc-355b2f7f6fa4-3_757_810_1050_680} The diagram shows a circle which passes through the points \(A ( 2,9 )\) and \(B ( 10,3 ) . A B\) is a diameter of the circle.

1. Calculate the radius of the circle and the coordinates of the centre.
2. Show that the equation of the circle may be written in the form \(x ^ { 2 } + y ^ { 2 } - 12 x - 12 y + 47 = 0\).
3. The tangent to the circle at the point \(B\) cuts the \(x\)-axis at \(C\). Find the coordinates of \(C\).

12 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 4 y = 9\).

1. Show that the centre of this circle is \(\mathrm { C } ( 4,2 )\) and find the radius of the circle.
2. Show that the origin lies inside the circle.
3. Show that AB is a diameter of the circle, where A has coordinates (2, 7) and B has coordinates \(( 6 , - 3 )\).
4. Find the equation of the tangent to the circle at A . Give your answer in the form \(y = m x + c\).