1.03b Straight lines: parallel and perpendicular relationships

10

1. The coordinates of two points \(A\) and \(B\) are \(( - 7,3 )\) and \(( 5,11 )\) respectively.
   Show that the equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = 11\).
2. A circle passes through \(A\) and \(B\) and its centre lies on the line \(12 x - 5 y = 70\). Find an equation of the circle.

6 Points \(A\) and \(B\) have coordinates \(( 8,3 )\) and \(( p , q )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(y = - 2 x + 4\). Find the values of \(p\) and \(q\).

7 The point \(A\) has coordinates \(( 1,5 )\) and the line \(l\) has gradient \(- \frac { 2 } { 3 }\) and passes through \(A\). A circle has centre \(( 5,11 )\) and radius \(\sqrt { 52 }\).

1. Show that \(l\) is the tangent to the circle at \(A\).
2. Find the equation of the other circle of radius \(\sqrt { 52 }\) for which \(l\) is also the tangent at \(A\).

11 The point \(P\) lies on the line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. A curve has equation \(y = - \frac { m } { x }\). There is a single point \(P\) on the curve such that the straight line is a tangent to the curve at \(P\).

1. Find the coordinates of \(P\), giving the \(y\)-coordinate in terms of \(m\).
   The normal to the curve at \(P\) intersects the curve again at the point \(Q\).
2. Find the coordinates of \(Q\) in terms of \(m\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-14_631_689_274_721} The diagram shows part of the curve with equation \(y = \frac { 4 } { ( 2 x - 1 ) ^ { 2 } }\) and parts of the lines \(x = 1\) and \(y = 1\). The curve passes through the points \(A ( 1,4 )\) and \(B , \left( \frac { 3 } { 2 } , 1 \right)\).

1. Find the exact volume generated when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
2. A triangle is formed from the tangent to the curve at \(B\), the normal to the curve at \(B\) and the \(x\)-axis. Find the area of this triangle.

9 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-12_583_661_260_740} The diagram shows a circle with centre \(A\) passing through the point \(B\). A second circle has centre \(B\) and passes through \(A\). The tangent at \(B\) to the first circle intersects the second circle at \(C\) and \(D\). The coordinates of \(A\) are ( \(- 1,4\) ) and the coordinates of \(B\) are ( 3,2 ).

1. Find the equation of the tangent CBD.
2. Find an equation of the circle with centre \(B\).
3. Find, by calculation, the \(x\)-coordinates of \(C\) and \(D\). \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-14_746_652_262_744} The diagram shows a sector \(C A B\) which is part of a circle with centre \(C\). A circle with centre \(O\) and radius \(r\) lies within the sector and touches it at \(D , E\) and \(F\), where \(C O D\) is a straight line and angle \(A C D\) is \(\theta\) radians.

1 Points \(A\) and \(B\) have coordinates \(( 5,2 )\) and \(( 10 , - 1 )\) respectively.

1. Find the equation of the perpendicular bisector of \(A B\).
2. Find the equation of the circle with centre \(A\) which passes through \(B\).

11 The coordinates of points \(A , B\) and \(C\) are \(A ( 5 , - 2 ) , B ( 10,3 )\) and \(C ( 2 p , p )\), where \(p\) is a constant.

1. Given that \(A C\) and \(B C\) are equal in length, find the value of the fraction \(p\).
2. It is now given instead that \(A C\) is perpendicular to \(B C\) and that \(p\) is an integer.
   1. Find the value of \(p\).
   2. Find the equation of the circle which passes through \(A , B\) and \(C\), giving your answer in the form \(x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0\), where \(a , b\) and \(c\) are constants.
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 The circle \(x ^ { 2 } + y ^ { 2 } + 4 x - 2 y - 20 = 0\) has centre \(C\) and passes through points \(A\) and \(B\).

1. State the coordinates of \(C\).
   It is given that the midpoint, \(D\), of \(A B\) has coordinates \(\left( 1 \frac { 1 } { 2 } , 1 \frac { 1 } { 2 } \right)\).
2. Find the equation of \(A B\), giving your answer in the form \(y = m x + c\).
3. Find, by calculation, the \(x\)-coordinates of \(A\) and \(B\).

7 The line \(L _ { 1 }\) has equation \(2 x + y = 8\). The line \(L _ { 2 }\) passes through the point \(A ( 7,4 )\) and is perpendicular to \(L _ { 1 }\).

1. Find the equation of \(L _ { 2 }\).
2. Given that the lines \(L _ { 1 }\) and \(L _ { 2 }\) intersect at the point \(B\), find the length of \(A B\).

6 The curve \(y = 9 - \frac { 6 } { x }\) and the line \(y + x = 8\) intersect at two points. Find

1. the coordinates of the two points,
2. the equation of the perpendicular bisector of the line joining the two points.

6 \includegraphics[max width=\textwidth, alt={}, center]{b24ed4c7-ab07-45f4-adf2-027734c36b62-3_593_878_269_635} The diagram shows a rectangle \(A B C D\). The point \(A\) is \(( 2,14 ) , B\) is \(( - 2,8 )\) and \(C\) lies on the \(x\)-axis. Find

1. the equation of \(B C\),
2. the coordinates of \(C\) and \(D\).

11 \includegraphics[max width=\textwidth, alt={}, center]{d71002bb-b6f0-42a3-89fb-f2769d5c3779-4_563_965_813_591} In the diagram, the points \(A\) and \(C\) lie on the \(x\) - and \(y\)-axes respectively and the equation of \(A C\) is \(2 y + x = 16\). The point \(B\) has coordinates ( 2,2 ). The perpendicular from \(B\) to \(A C\) meets \(A C\) at the point \(X\).

1. Find the coordinates of \(X\). The point \(D\) is such that the quadrilateral \(A B C D\) has \(A C\) as a line of symmetry.
2. Find the coordinates of \(D\).
3. Find, correct to 1 decimal place, the perimeter of \(A B C D\).

8 \includegraphics[max width=\textwidth, alt={}, center]{3b527397-7781-41e9-8218-57277cc977bf-3_599_716_1071_717} The diagram shows points \(A , B\) and \(C\) lying on the line \(2 y = x + 4\). The point \(A\) lies on the \(y\)-axis and \(A B = B C\). The line from \(D ( 10 , - 3 )\) to \(B\) is perpendicular to \(A C\). Calculate the coordinates of \(B\) and \(C\).

8 \includegraphics[max width=\textwidth, alt={}, center]{56d4d40a-32f5-4f2d-938e-a24312cd42e7-3_625_547_1489_797} The diagram shows a triangle \(A B C\) in which \(A\) is \(( 3 , - 2 )\) and \(B\) is \(( 15,22 )\). The gradients of \(A B , A C\) and \(B C\) are \(2 m , - 2 m\) and \(m\) respectively, where \(m\) is a positive constant.

1. Find the gradient of \(A B\) and deduce the value of \(m\).
2. Find the coordinates of \(C\). The perpendicular bisector of \(A B\) meets \(B C\) at \(D\).
3. Find the coordinates of \(D\).

8 \includegraphics[max width=\textwidth, alt={}, center]{71fe6352-e0dc-4c3a-8b54-99709a1782ca-3_796_695_1539_726} The diagram shows a rhombus \(A B C D\) in which the point \(A\) is ( \(- 1,2\) ), the point \(C\) is ( 5,4 ) and the point \(B\) lies on the \(y\)-axis. Find

1. the equation of the perpendicular bisector of \(A C\),
2. the coordinates of \(B\) and \(D\),
3. the area of the rhombus.

9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).

1. Find the equation of \(M B\) and the coordinates of \(B\).
2. Show that \(A B\) is perpendicular to \(B C\).
3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).

7 The point \(R\) is the reflection of the point \(( - 1,3 )\) in the line \(3 y + 2 x = 33\). Find by calculation the coordinates of \(R\).

7 \includegraphics[max width=\textwidth, alt={}, center]{13cfb59a-7781-4786-a625-919b01a2a4f0-3_465_554_255_794} The diagram shows three points \(A ( 2,14 ) , B ( 14,6 )\) and \(C ( 7,2 )\). The point \(X\) lies on \(A B\), and \(C X\) is perpendicular to \(A B\). Find, by calculation,

1. the coordinates of \(X\),
2. the ratio \(A X : X B\).

1 Find the coordinates of the point at which the perpendicular bisector of the line joining (2, 7) to \(( 10,3 )\) meets the \(x\)-axis.

11 \includegraphics[max width=\textwidth, alt={}, center]{0b047754-84f2-46ea-b441-7c68cef47641-4_995_867_260_639} The diagram shows a parallelogram \(A B C D\), in which the equation of \(A B\) is \(y = 3 x\) and the equation of \(A D\) is \(4 y = x + 11\). The diagonals \(A C\) and \(B D\) meet at the point \(E \left( 6 \frac { 1 } { 2 } , 8 \frac { 1 } { 2 } \right)\). Find, by calculation, the coordinates of \(A , B , C\) and \(D\).

7 The point \(A\) has coordinates \(( p , 1 )\) and the point \(B\) has coordinates \(( 9,3 p + 1 )\), where \(p\) is a constant.

1. For the case where the distance \(A B\) is 13 units, find the possible values of \(p\).
2. For the case in which the line with equation \(2 x + 3 y = 9\) is perpendicular to \(A B\), find the value of \(p\).

8 Three points have coordinates \(A ( 0,7 ) , B ( 8,3 )\) and \(C ( 3 k , k )\). Find the value of the constant \(k\) for which

1. \(C\) lies on the line that passes through \(A\) and \(B\),
2. \(C\) lies on the perpendicular bisector of \(A B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{8c358a10-a3e1-47b5-ae62-30ba6b76c167-3_655_1011_255_566} The diagram shows triangle \(A B C\) where \(A B = 5 \mathrm {~cm} , A C = 4 \mathrm {~cm}\) and \(B C = 3 \mathrm {~cm}\). Three circles with centres at \(A , B\) and \(C\) have radii \(3 \mathrm {~cm} , 2 \mathrm {~cm}\) and 1 cm respectively. The circles touch each other at points \(E , F\) and \(G\), lying on \(A B , A C\) and \(B C\) respectively. Find the area of the shaded region \(E F G\).

2 The point \(A\) has coordinates ( \(- 2,6\) ). The equation of the perpendicular bisector of the line \(A B\) is \(2 y = 3 x + 5\).

1. Find the equation of \(A B\).
2. Find the coordinates of \(B\).