1.03e Complete the square: find centre and radius of circle

1 \begin{figure}[h] \end{figure} Fig. 11 shows the line through the points \(\mathrm { A } ( - 1,3 )\) and \(\mathrm { B } ( 5,1 )\).

1. Find the equation of the line through \(\mathbf { A }\) and \(\mathbf { B }\).
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac { 32 } { 3 }\) square units.
3. Show that the equation of the perpendicular bisector of AB is \(y = 3 x - 4\).
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle.

7

1. Find the equation of the line passing through \(\mathrm { A } ( - 1,1 )\) and \(\mathrm { B } ( 3,9 )\).
2. Show that the equation of the perpendicular bisector of AB is \(2 y + x = 11\).
3. A circle has centre \(( 5,3 )\), so that its equation is \(( x - 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = k\). Given that the circle passes through A , show that \(k = 40\). Show that the circle also passes through B .
4. Find the \(x\)-coordinates of the points where this circle crosses the \(x\)-axis. Give your answers in surd form.

7.A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) .
A line \(l\) has equation \(y = m x + c\)

1. Show that the \(x\) coordinates of the points where \(C\) and \(l\) intersect satisfy $$\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 2 ( a - m ( c - b ) ) x + a ^ { 2 } + ( c - b ) ^ { 2 } - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\) ,
2. show that $$c = b - m a \pm r \sqrt { m ^ { 2 } + 1 }$$ The circle \(C _ { 1 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 16 = 0$$ and the circle \(C _ { 2 }\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$
3. Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ,justifying your answer.
4. Find the equations of all lines that are a tangent to both \(C _ { 1 }\) and \(C _ { 2 }\) [You may find the following Pythagorean triple helpful in this part: \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ]

1.A point \(P\) lies on the curve with equation $$x ^ { 2 } + y ^ { 2 } - 6 x + 8 y = 24$$ Find the greatest and least possible values of the length \(O P\) ,where \(O\) is the origin.

4.The line with equation \(y = m x\) is a tangent to the circle \(C _ { 1 }\) with equation $$( x + 4 ) ^ { 2 } + ( y - 7 ) ^ { 2 } = 13$$

1. Show that \(m\) satisfies the equation $$3 m ^ { 2 } + 56 m + 36 = 0$$ The tangents from the origin \(O\) to \(C _ { 1 }\) touch \(C _ { 1 }\) at the points \(A\) and \(B\) .
2. Find the coordinates of the points \(A\) and \(B\) .
   (8)
   Another circle \(C _ { 2 }\) has equation \(x ^ { 2 } + y ^ { 2 } = 13\) .The tangents from the point \(( 4 , - 7 )\) to \(C _ { 2 }\) touch it at the points \(P\) and \(Q\) .
3. Find the coordinates of either the point \(P\) or the point \(Q\) .
   (2)

7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).

1. Find the value of \(k\).
2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
3. Write down the coordinates of the centre and the radius of the circle.
4. Verify that \(A B\) is a diameter of the circle.

8 A circle has equation \(x ^ { 2 } + y ^ { 2 } + 6 x - 4 y - 4 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of the points where the circle meets the line with equation \(y = 3 x + 4\).

9 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 2 y - 3 = 0\).

1. Find the coordinates of \(C\) and the radius of the circle.
2. Find the values of \(k\) for which the line \(y = k\) is a tangent to the circle, giving your answers in simplified surd form.
3. The points \(S\) and \(T\) lie on the circumference of the circle. \(M\) is the mid-point of the chord \(S T\). Given that the length of \(C M\) is 2 , calculate the length of the chord \(S T\).
4. Find the coordinates of the point where the circle meets the line \(x - 2 y - 12 = 0\).

10 A circle has centre \(C ( - 2,4 )\) and radius 5 .

1. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
2. Show that the tangent to the circle at the point \(P ( - 5,8 )\) has equation \(3 x - 4 y + 47 = 0\).
3. Verify that the point \(T ( 3,14 )\) lies on this tangent.
4. Find the area of the triangle \(C P T\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

6

1. Sketch the curve \(y = - \sqrt { x }\).
2. Describe fully a transformation that transforms the curve \(y = - \sqrt { x }\) to the curve \(y = 5 - \sqrt { x }\).
3. The curve \(y = - \sqrt { x }\) is stretched by a scale factor of 2 parallel to the \(x\)-axis. State the equation of the curve after it has been stretched.

9

1. The line joining the points \(A ( 4,5 )\) and \(B ( p , q )\) has mid-point \(M ( - 1,3 )\). Find \(p\) and \(q\). \(A B\) is the diameter of a circle.
2. Find the radius of the circle.
3. Find the equation of the circle, giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).
4. Find an equation of the tangent to the circle at the point \(( 4,5 )\).

9 The points \(A ( 1,3 ) , B ( 7,1 )\) and \(C ( - 3 , - 9 )\) are joined to form a triangle.

1. Show that this triangle is right-angled and state whether the right angle is at \(A , B\) or \(C\).
2. The points \(A , B\) and \(C\) lie on the circumference of a circle. Find the equation of the circle in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\).

10 A circle has equation \(( x - 5 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).

1. Find the coordinates of the centre \(C\) and the length of the diameter.
2. Find the equation of the line which passes through \(C\) and the point \(P ( 7,2 )\).
3. Calculate the length of \(C P\) and hence determine whether \(P\) lies inside or outside the circle.
4. Determine algebraically whether the line with equation \(y = 2 x\) meets the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

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

1. Find the coordinates of \(C\) and the radius of the circle.
2. Show that the tangent to the circle at the point \(P ( 8,2 )\) has equation \(3 x + 4 y = 32\).
3. The circle meets the \(y\)-axis at \(Q\) and the tangent meets the \(y\)-axis at \(R\). Find the area of triangle \(P Q R\).

10 \includegraphics[max width=\textwidth, alt={}, center]{0ae3af7e-32cc-43fa-89bb-d6697a8f8061-3_755_905_248_580} The diagram shows the circle with equation \(x ^ { 2 } + y ^ { 2 } - 8 x - 6 y - 20 = 0\).

1. Find the centre and radius of the circle. The circle crosses the positive \(x\)-axis at the point \(A\).
2. Find the equation of the tangent to the circle at \(A\).
3. A second tangent to the circle is parallel to the tangent at \(A\). Find the equation of this second tangent.
4. Another circle has centre at the origin \(O\) and radius \(r\). This circle lies wholly inside the first circle. Find the set of possible values of \(r\).

12 A circle has equation \(( x - 2 ) ^ { 2 } + y ^ { 2 } = 20\).

1. Write down the radius of the circle and the coordinates of its centre.
2. Find the points of intersection of the circle with the \(y\)-axis and sketch the circle.
3. Show that, where the line \(y = 2 x + k\) intersects the circle, $$5 x ^ { 2 } + ( 4 k - 4 ) x + k ^ { 2 } - 16 = 0 .$$
4. Hence find the values of \(k\) for which the line \(y = 2 x + k\) is a tangent to the circle. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

10

1. Points A and B have coordinates \(( - 2,1 )\) and \(( 3,4 )\) respectively. Find the equation of the perpendicular bisector of AB and show that it may be written as \(5 x + 3 y = 10\).
2. Points C and D have coordinates \(( - 5,4 )\) and \(( 3,6 )\) respectively. The line through C and D has equation \(4 y = x + 21\). The point E is the intersection of CD and the perpendicular bisector of AB . Find the coordinates of point E .
3. Find the equation of the circle with centre E which passes through A and B . Show also that CD is a diameter of this circle.

10 Fig. 10 shows a sketch of a circle with centre \(\mathrm { C } ( 4,2 )\). The circle intersects the \(x\)-axis at \(\mathrm { A } ( 1,0 )\) and at B . \begin{figure}[h] \end{figure}

1. Write down the coordinates of B .
2. Find the radius of the circle and hence write down the equation of the circle.
3. AD is a diameter of the circle. Find the coordinates of D .
4. Find the equation of the tangent to the circle at D . Give your answer in the form \(y = a x + b\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-4_643_853_269_589} \captionsetup{labelformat=empty} \caption{Fig. 11}
   \end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
5. Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
6. Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
7. The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
8. Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2e8f2d63-8a25-4da2-8c3e-9e75ea1b7c08-5_775_1461_317_296} \captionsetup{labelformat=empty} \caption{Fig. 12}
   \end{figure} Fig. 12 shows the graph of a cubic curve. It intersects the axes at \(( - 5,0 ) , ( - 2,0 ) , ( 1.5,0 )\) and \(( 0 , - 30 )\).
9. Use the intersections with both axes to express the equation of the curve in a factorised form.
10. Hence show that the equation of the curve may be written as \(y = 2 x ^ { 3 } + 11 x ^ { 2 } - x - 30\).
11. Draw the line \(y = 5 x + 10\) accurately on the graph. The curve and this line intersect at ( \(- 2,0\) ); find graphically the \(x\)-coordinates of the other points of intersection.
12. Show algebraically that the \(x\)-coordinates of the other points of intersection satisfy the equation $$2 x ^ { 2 } + 7 x - 20 = 0 .$$ Hence find the exact values of the \(x\)-coordinates of the other points of intersection. \section*{END OF QUESTION PAPER}

3 The diagram shows a sketch of a circle which passes through the origin \(O\). \includegraphics[max width=\textwidth, alt={}, center]{13cfb9ca-9495-4b69-80c5-9fb7e8e0f957-3_423_451_356_794} The equation of the circle is \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\) and \(O A\) is a diameter.

1. Find the cartesian coordinates of the point \(A\).
2. Using \(O\) as the pole and the positive \(x\)-axis as the initial line, the polar coordinates of \(A\) are \(( k , \alpha )\).
   1. Find the value of \(k\) and the value of \(\tan \alpha\).
   2. Find the polar equation of the circle \(( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 25\), giving your answer in the form \(r = p \cos \theta + q \sin \theta\).

6 The polar equation of a curve \(C _ { 1 }\) is $$r = 2 ( \cos \theta - \sin \theta ) , \quad 0 \leqslant \theta \leqslant 2 \pi$$

1. 1. Find the cartesian equation of \(C _ { 1 }\).
   2. Deduce that \(C _ { 1 }\) is a circle and find its radius and the cartesian coordinates of its centre.
2. The diagram shows the curve \(C _ { 2 }\) with polar equation $$r = 4 + \sin \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ \includegraphics[max width=\textwidth, alt={}, center]{90a59b47-3799-46a2-b76b-ced5cc3e1aac-4_519_847_443_593}
   1. Find the area of the region that is bounded by \(C _ { 2 }\).
   2. Prove that the curves \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
   3. Find the area of the region that is outside \(C _ { 1 }\) but inside \(C _ { 2 }\).

5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

1. Find the centre and radius of the circle.
2. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
3. State what can be deduced from the answer to part (ii) about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).

3

1. A circle is defined by the parametric equations \(x = 3 + 2 \cos \theta , y = - 4 + 2 \sin \theta\).
   1. Find a cartesian equation of the circle.
   2. Write down the centre and radius of the circle.
2. In this question you must show detailed reasoning. The curve \(S\) is defined by the parametric equations \(x = 4 \cos t , y = 2 \sin t\). The line \(L\) is a tangent to \(S\) at the point given by \(t = \frac { 1 } { 6 } \pi\). Find where the line \(L\) cuts the \(x\)-axis.

1 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } + 8 x - 2 y - 7 = 0\).
Find

1. the coordinates of \(C\),
2. the radius of the circle.

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0$$

1. Find
   1. the coordinates of the centre of \(C\)
   2. the radius of \(C\) The line with equation \(y = k x\), where \(k\) is a constant, cuts \(C\) at two distinct points.
2. Find the range of values for \(k\).

1. A circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 4 x + 8 y - 8 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\). The straight line with equation \(x = k\), where \(k\) is a constant, is a tangent to \(C\).
2. Find the possible values for \(k\).