1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

3 The line \(\frac { x } { a } + \frac { y } { b } = 1\), where \(a\) and \(b\) are positive constants, meets the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Given that \(P Q = \sqrt { } ( 45 )\) and that the gradient of the line \(P Q\) is \(- \frac { 1 } { 2 }\), find the values of \(a\) and \(b\).

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 \(A X : X B\).

7 The coordinates of points \(A\) and \(B\) are \(( a , 2 )\) and \(( 3 , b )\) respectively, where \(a\) and \(b\) are constants. The distance \(A B\) is \(\sqrt { } ( 125 )\) units and the gradient of the line \(A B\) is 2 . Find the possible values of \(a\) and of \(b\).

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 \(D\).

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 \(E F G\).

11 Triangle \(A B C\) has vertices at \(A ( - 2 , - 1 ) , B ( 4,6 )\) and \(C ( 6 , - 3 )\).

1. Show that triangle \(A B C\) is isosceles and find the exact area of this triangle.
2. The point \(D\) is the point on \(A B\) such that \(C D\) is perpendicular to \(A B\). Calculate the \(x\)-coordinate of \(D\).

5 The equation of a curve is \(y = 2 \cos x\).

1. Sketch the graph of \(y = 2 \cos x\) for \(- \pi \leqslant x \leqslant \pi\), stating the coordinates of the point of intersection with the \(y\)-axis. Points \(P\) and \(Q\) lie on the curve and have \(x\)-coordinates of \(\frac { 1 } { 3 } \pi\) and \(\pi\) respectively.
2. Find the length of \(P Q\) correct to 1 decimal place.
   The line through \(P\) and \(Q\) meets the \(x\)-axis at \(H ( h , 0 )\) and the \(y\)-axis at \(K ( 0 , k )\).
3. Show that \(h = \frac { 5 } { 9 } \pi\) and find the value of \(k\).

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 \(B\).

5 \includegraphics[max width=\textwidth, alt={}, center]{5df7bd9f-31cc-41a3-b1c0-3ee9366e6d8a-08_558_785_258_680} The diagram shows a kite \(O A B C\) in which \(A C\) is the line of symmetry. The coordinates of \(A\) and \(C\) are \(( 0,4 )\) and \(( 8,0 )\) respectively and \(O\) is the origin.

1. Find the equations of \(A C\) and \(O B\).
2. Find, by calculation, the coordinates of \(B\).

8 Points \(A\) and \(B\) have coordinates \(( h , h )\) and \(( 4 h + 6,5 h )\) respectively. The equation of the perpendicular bisector of \(A B\) is \(3 x + 2 y = k\). Find the values of the constants \(h\) and \(k\).

6 The coordinates of points \(A\) and \(B\) are \(( - 3 k - 1 , k + 3 )\) and \(( k + 3,3 k + 5 )\) respectively, where \(k\) is a constant ( \(k \neq - 1\) ).

1. Find and simplify the gradient of \(A B\), showing that it is independent of \(k\).
2. Find and simplify the equation of the perpendicular bisector of \(A B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-06_625_750_260_699} The diagram shows a trapezium \(A B C D\) in which the coordinates of \(A , B\) and \(C\) are (4, 0), (0, 2) and \(( h , 3 h )\) respectively. The lines \(B C\) and \(A D\) are parallel, angle \(A B C = 90 ^ { \circ }\) and \(C D\) is parallel to the \(x\)-axis.

1. Find, by calculation, the value of \(h\).
2. Hence find the coordinates of \(D\).

2 Two points \(A\) and \(B\) have coordinates \(( 1,3 )\) and \(( 9 , - 1 )\) respectively. The perpendicular bisector of \(A B\) intersects the \(y\)-axis at the point \(C\). Find the coordinates of \(C\).

6 The equation of a curve is \(y = 3 \cos 2 x\) and the equation of a line is \(2 y + \frac { 3 x } { \pi } = 5\).

1. State the smallest and largest values of \(y\) for both the curve and the line for \(0 \leqslant x \leqslant 2 \pi\).
2. Sketch, on the same diagram, the graphs of \(y = 3 \cos 2 x\) and \(2 y + \frac { 3 x } { \pi } = 5\) for \(0 \leqslant x \leqslant 2 \pi\).
3. State the number of solutions of the equation \(6 \cos 2 x = 5 - \frac { 3 x } { \pi }\) for \(0 \leqslant x \leqslant 2 \pi\).

5 Two points have coordinates \(A ( 5,7 )\) and \(B ( 9 , - 1 )\).

1. Find the equation of the perpendicular bisector of \(A B\). The line through \(C ( 1,2 )\) parallel to \(A B\) meets the perpendicular bisector of \(A B\) at the point \(X\).
2. Find, by calculation, the distance \(B X\).

9 \includegraphics[max width=\textwidth, alt={}, center]{a10ad459-6f86-4845-ba28-e4e394bf3d1e-4_719_958_264_589} The diagram shows a rectangle \(A B C D\), where \(A\) is \(( 3,2 )\) and \(B\) is \(( 1,6 )\).

1. Find the equation of \(B C\). Given that the equation of \(A C\) is \(y = x - 1\), find
2. the coordinates of \(C\),
3. the perimeter of the rectangle \(A B C D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1cf37a58-8a7f-4dc8-9e35-2e8badf3eb83-2_594_778_1360_682} The diagram shows a trapezium \(A B C D\) in which \(B C\) is parallel to \(A D\) and angle \(B C D = 90 ^ { \circ }\). The coordinates of \(A , B\) and \(D\) are \(( 2,0 ) , ( 4,6 )\) and \(( 12,5 )\) respectively.

1. Find the equations of \(B C\) and \(C D\).
2. Calculate the coordinates of \(C\).

7 Three points have coordinates \(A ( 2,6 ) , B ( 8,10 )\) and \(C ( 6,0 )\). The perpendicular bisector of \(A B\) meets the line \(B C\) at \(D\). Find

1. the equation of the perpendicular bisector of \(A B\) in the form \(a x + b y = c\),
2. the coordinates of \(D\).

5 \includegraphics[max width=\textwidth, alt={}, center]{dd2cb0ec-5df9-4d99-9e15-5ae1f1c07b96-3_684_771_260_685} The three points \(A ( 1,3 ) , B ( 13,11 )\) and \(C ( 6,15 )\) are shown in the diagram. The perpendicular from \(C\) to \(A B\) meets \(A B\) at the point \(D\). Find

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

6 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-2_627_748_1685_699} The three points \(A ( 3,8 ) , B ( 6,2 )\) and \(C ( 10,2 )\) are shown in the diagram. The point \(D\) is such that the line \(D A\) is perpendicular to \(A B\) and \(D C\) is parallel to \(A B\). Calculate the coordinates of \(D\).

2 The equation of a curve is \(y = 3 \cos 2 x\). The equation of a line is \(x + 2 y = \pi\). On the same diagram, sketch the curve and the line for \(0 \leqslant x \leqslant \pi\).