Tangent condition using discriminant

10 The equation of a circle is \(( x - a ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). The line \(y = \frac { 1 } { 2 } x + 6\) is a tangent to the circle at the point \(P\).

1. Show that one possible value of \(a\) is 4 and find the other possible value.
2. For \(a = 4\), find the equation of the normal to the circle at \(P\).
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b).

7 The equation of a circle is \(( x - 6 ) ^ { 2 } + ( y + a ) ^ { 2 } = 18\). The line with equation \(y = 2 a - x\) is a tangent to the circle.

1. Find the two possible values of the constant \(a\).
2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent.
   \includegraphics[max width=\textwidth, alt={}, center]{84b3cd80-faf0-4522-8286-52bf7f86dc8a-14_280_1358_317_349} The diagram shows a symmetrical plate \(A B C D E F\). The line \(A B C D\) is straight and the length of \(B C\) is 2 cm . Each of the two sectors \(A B F\) and \(D C E\) is of radius \(r \mathrm {~cm}\) and each of the angles \(A B F\) and \(D C E\) is equal to \(\frac { 1 } { 3 } \pi\) radians.
3. It is given that \(r = 0.4 \mathrm {~cm}\).
   1. Show that the length \(\mathrm { EF } = 2.4 \mathrm {~cm}\).
   2. Find the area of the plate. Give your answer correct to 3 significant figures.
4. It is given instead that the perimeter of the plate is 6 cm . Find the value of \(r\). Give your answer correct to 3 significant figures.

8 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + p x + 2 y + q = 0\), where \(p\) and \(q\) are constants.

1. Express the equation in the form \(( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }\), where \(a\) is to be given in terms of \(p\) and \(r ^ { 2 }\) is to be given in terms of \(p\) and \(q\).
   The line with equation \(x + 2 y = 10\) is the tangent to the circle at the point \(A ( 4,3 )\).
2. 1. Find the equation of the normal to the circle at the point \(A\).
   2. Find the values of \(p\) and \(q\).

9. A circle \(C\) has equation $$( x - k ) ^ { 2 } + ( y - 2 k ) ^ { 2 } = k + 7$$ where \(k\) is a positive constant.

1. Write down, in terms of \(k\),
   1. the coordinates of the centre of \(C\),
   2. the radius of \(C\). Given that the point \(P ( 2,3 )\) lies on \(C\)
2. 1. show that \(5 k ^ { 2 } - 17 k + 6 = 0\)
   2. hence find the possible values of \(k\). The tangent to the circle at \(P\) intersects the \(x\)-axis at point \(T\).
      Given that \(k < 2\)
3. calculate the exact area of triangle \(O P T\).

10. The circle \(C\) has centre \(X ( 3,5 )\) and radius \(r\) The line \(l\) has equation \(y = 2 x + k\), where \(k\) is a constant.

1. Show that \(l\) and \(C\) intersect when $$5 x ^ { 2 } + ( 4 k - 26 ) x + k ^ { 2 } - 10 k + 34 - r ^ { 2 } = 0$$ Given that \(l\) is a tangent to \(C\),
2. show that \(5 r ^ { 2 } = ( k + p ) ^ { 2 }\), where \(p\) is a constant to be found. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db4ec300-8081-4d29-acd5-0aae789d8f95-28_636_572_902_687} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} The line \(l\)
   • cuts the \(y\)-axis at the point \(A\)
   • touches the circle \(C\) at the point \(B\)
     as shown in Figure 2.
     Given that \(A B = 2 r\)
   • find the value of \(k\)
     

11 \begin{figure}[h] \end{figure} A circle has centre \(C ( 1,3 )\) and passes through the point \(A ( 3,7 )\) as shown in Fig. 11.

1. Show that the equation of the tangent at A is \(x + 2 y = 17\).
2. The line with equation \(y = 2 x - 9\) intersects this tangent at the point T . Find the coordinates of T .
3. The equation of the circle is \(( x - 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 20\). Show that the line with equation \(y = 2 x - 9\) is a tangent to the circle. Give the coordinates of the point where this tangent touches the circle.

7. A circle has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of the circle.
2. Find an equation for the circle.
3. Show that the line \(y = 2 x - 3\) is a tangent to the circle and find the coordinates of the point of contact.

5 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.

4 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).

1. State the coordinates of the centre and the radius of this circle.
2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.

7．A circle \(C\) has centre \(X ( a , b )\) and radius \(r\) ．
A line \(l\) has equation \(y = m x + c\)
（a）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 \(C\) ，
（b）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 $$x ^ { 2 } + y ^ { 2 } - 20 x - 10 y + 89 = 0$$ （c）Find the equations of any lines that are normal to both \(C _ { 1 }\) and \(C _ { 2 }\) ，justifying your answer．
（d）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： \(7 ^ { 2 } + 24 ^ { 2 } = 25 ^ { 2 }\) ］

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.}

13 A circle has equation \(( x - 5 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 20\).

1. State the coordinates of the centre and the radius of this circle.
2. State, with a reason, whether or not this circle intersects the \(y\)-axis.
3. Find the equation of the line parallel to the line \(y = 2 x\) that passes through the centre of the circle.
4. Show that the line \(y = 2 x + 2\) is a tangent to the circle. State the coordinates of the point of contact.

4 A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + k = 0\).

1. Find the set of possible values of \(k\).
2. It is given that \(k = - 46\). Determine the coordinates of the two points on \(C\) at which the gradient of the tangent is \(\frac { 1 } { 2 }\).

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 \(k\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } - 10 x + 4 y + 11 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact radius of \(C\), giving your answer as a simplified surd. The line \(l\) has equation \(y = 3 x + k\) where \(k\) is a constant.
      Given that \(l\) is a tangent to \(C\),
2. find the possible values of \(k\), giving your answers as simplified surds.

7 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + 9 = 0\).

1. Express this equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle.
3. The point \(D\) has coordinates (7, -2).
   1. Verify that the point \(D\) lies on the circle.
   2. Find an equation of the normal to the circle at the point \(D\), giving your answer in the form \(m x + n y = p\), where \(m , n\) and \(p\) are integers.
4. 1. A line has equation \(y = k x\). Show that the \(x\)-coordinates of any points of intersection of the line and the circle satisfy the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$
   2. Find the values of \(k\) for which the equation $$\left( k ^ { 2 } + 1 \right) x ^ { 2 } + 2 ( 5 k - 3 ) x + 9 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (d)(ii).

7 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 4 x - 14 = 0\).

1. Find:
   1. the coordinates of the centre of the circle;
   2. the radius of the circle in the form \(p \sqrt { 2 }\), where \(p\) is an integer.
2. A chord of the circle has length 8. Find the perpendicular distance from the centre of the circle to this chord.
3. A line has equation \(y = 2 k - x\), where \(k\) is a constant.
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$
   2. Find the values of \(k\) for which the equation $$x ^ { 2 } - 2 ( k + 1 ) x + 2 k ^ { 2 } - 7 = 0$$ has equal roots.
   3. Describe the geometrical relationship between the line and the circle when \(k\) takes either of the values found in part (c)(ii).

7. The circle \(C\) has centre \(( 5,2 )\) and passes through the point \(( 7,3 )\).

1. Find the length of the diameter of \(C\).
2. Find an equation for \(C\).
3. Show that the line \(y = 2 x - 3\) is a tangent to \(C\) and find the coordinates of the point of contact.

8 In this question you must show detailed reasoning. The centre of a circle C is at the point \(( - 1,3 )\) and C passes through the point \(( 1 , - 1 )\). The straight line L passes through the points \(( 1,9 )\) and \(( 4,3 )\). Show that L is a tangent to C .

4 A circle with centre \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 10 y + 20 = 0\).

1. By completing the square, express this equation in the form $$x ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Write down:
   1. the coordinates of \(C\);
   2. the radius of the circle, leaving your answer in surd form.
3. A line has equation \(y = 2 x\).
   1. Show that the \(x\)-coordinate of any point of intersection of the line and the circle satisfies the equation \(x ^ { 2 } - 4 x + 4 = 0\).
   2. Hence show that the line is a tangent to the circle and find the coordinates of the point of contact, \(P\).
4. Prove that the point \(Q ( - 1,4 )\) lies inside the circle.