1.03b Straight lines: parallel and perpendicular relationships

2 The point \(M\) is the mid-point of the line joining the points \(( 3,7 )\) and \(( - 1,1 )\). Find the equation of the line through \(M\) which is parallel to the line \(\frac { x } { 3 } + \frac { y } { 2 } = 1\).

5 \includegraphics[max width=\textwidth, alt={}, center]{e835a60b-fbeb-49fb-ba6b-ac12c702d487-08_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 \(B\).

6. The line \(l _ { 1 }\) has equation \(3 x - 4 y + 20 = 0\) The line \(l _ { 2 }\) cuts the \(x\)-axis at \(R ( 8,0 )\) and is parallel to \(l _ { 1 }\)

1. Find the equation of \(l _ { 2 }\), writing your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers to be found. The line \(l _ { 1 }\) cuts the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\).
   Given that \(P Q R S\) is a parallelogram, find
2. the area of \(P Q R S\),
3. the coordinates of \(S\).

8. The line \(l _ { 1 }\) has equation $$2 x - 5 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) Given that
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(M\).
2. Using algebra and showing all your working, find the coordinates of \(M\).
   (Solutions relying on calculator technology are not acceptable.) Given that the diagonals of a square \(A B C D\) meet at \(M\),
3. find the coordinates of the point \(C\).

5. \begin{figure}[h] \end{figure} \section*{In this question you must show all stages of your working.} \section*{Solutions relying on calculator technology are not acceptable.} The straight line \(l _ { 1 }\), shown in Figure 1, passes through the points \(P ( - 2,9 )\) and \(Q ( 10,6 )\).

1. Find the equation of \(l _ { 1 }\), giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found. The straight line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\) as shown in Figure 1.
2. Find the coordinates of \(R\)
3. Find the exact area of triangle \(O P Q\).

7. \begin{figure}[h] \end{figure} The line \(l _ { 1 }\) has equation \(4 y + 3 x = 48\) The line \(l _ { 1 }\) cuts the \(y\)-axis at the point \(C\), as shown in Figure 3.

1. State the \(y\) coordinate of \(C\). The point \(D ( 8,6 )\) lies on \(l _ { 1 }\) The line \(l _ { 2 }\) passes through \(D\) and is perpendicular to \(l _ { 1 }\) The line \(l _ { 2 }\) cuts the \(y\)-axis at the point \(E\) as shown in Figure 3.
2. Show that the \(y\) coordinate of \(E\) is \(- \frac { 14 } { 3 }\) A sector \(B C E\) of a circle with centre \(C\) is also shown in Figure 3. Given that angle \(B C E\) is 1.8 radians,
3. find the length of arc \(B E\). The region \(C B E D\), shown shaded in Figure 3, consists of the sector \(B C E\) joined to the triangle \(C D E\).
4. Calculate the exact area of the region \(C B E D\).

1. The curve \(C\) has equation \(y = \mathrm { f } ( x )\)

Given that

• \(\mathrm { f } ( x )\) is a quadratic expression
• the maximum turning point on \(C\) has coordinates \(( - 2,12 )\)
• \(C\) cuts the negative \(x\)-axis at - 5
  1. find \(\mathrm { f } ( x )\)

The line \(l _ { 1 }\) has equation \(y = \frac { 4 } { 5 } x\) Given that the line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through \(( - 5,0 )\)

find an equation for \(l _ { 2 }\), writing your answer in the form \(y = m x + c\) where \(m\) and \(c\) are constants to be found.
(3) \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-10_983_712_1126_616} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve \(C\) and the lines \(l _ { 1 }\) and \(l _ { 2 }\)

Define the region \(R\), shown shaded in Figure 2, using inequalities.

10. \begin{figure}[h] \end{figure} Figure 5 shows a sketch of the quadratic curve \(C\) with equation $$y = - \frac { 1 } { 4 } ( x + 2 ) ( x - b ) \quad \text { where } b \text { is a positive constant }$$ The line \(l _ { 1 }\) also shown in Figure 5,

• has gradient \(\frac { 1 } { 2 }\)
• intersects \(C\) on the negative \(x\)-axis and at the point \(P\)
• find, in terms of \(b\), an equation for \(l _ { 2 }\) Given also that \(l _ { 2 }\) intersects \(C\) at the point \(P\)
• show that another equation for \(l _ { 2 }\) is $$y = - 2 x + \frac { 5 b } { 2 } - 4$$
• Hence, or otherwise, find the value of \(b\)

1. Given that

• the point \(A\) has coordinates \(( 4,2 )\)
• the point \(B\) has coordinates \(( 15,7 )\)
• the line \(l _ { 1 }\) passes through \(A\) and \(B\)
  1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.

The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)

Find both possible pairs of coordinates of the point \(C\).

Hence find the minimum possible area of triangle \(A B C\).

6. The point \(A\) has coordinates \(( - 4,11 )\) and the point \(B\) has coordinates \(( 8,2 )\).

1. Find the gradient of the line \(A B\), giving your answer as a fully simplified fraction. The point \(M\) is the midpoint of \(A B\). The line \(l\) passes through \(M\) and is perpendicular to \(A B\).
2. Find an equation for \(l\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found. The point \(C\) lies on \(l\) such that the area of triangle \(A B C\) is 37.5 square units.
3. Find the two possible pairs of coordinates of point \(C\).

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4. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = \cos 2 x ^ { \circ } \quad 0 \leqslant x \leqslant k$$ The point \(Q\) and the point \(R ( k , 0 )\) lie on the curve and are shown in Figure 2.

1. State
   1. the coordinates of \(Q\),
   2. the value of \(k\).
2. Given that there are exactly two solutions to the equation $$\cos 2 x ^ { \circ } = p \quad \text { in the region } 0 \leqslant x \leqslant k$$ find the range of possible values for \(p\).

15. The points \(A\) and \(B\) have coordinates \(( - 8 , - 8 )\) and \(( 12,2 )\) respectively. \(A B\) is the diameter of a circle \(C\).

1. Find an equation for the circle \(C\). The point \(( 4,8 )\) also lies on \(C\).
2. Find an equation of the tangent to \(C\) at the point ( 4,8 ), giving your answer in the form \(a x + b y + c = 0\)

6. \begin{figure}[h] \end{figure} The straight line \(l _ { 1 }\) has equation \(2 y = 3 x + 5\) The line \(l _ { 1 }\) cuts the \(x\)-axis at the point \(A\), as shown in Figure 2.

1. 1. State the gradient of \(l _ { 1 }\)
   2. Write down the \(x\) coordinate of point \(A\). Another straight line \(l _ { 2 }\) intersects \(l _ { 1 }\) at the point \(B\) with \(x\) coordinate 1 and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
2. find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers,
3. find the exact area of triangle \(A B C\).

1. The line \(l _ { 1 }\) has equation \(x + 3 y - 11 = 0\)

The point \(A\) and the point \(B\) lie on \(l _ { 1 }\) Given that \(A\) has coordinates ( \(- 1 , p\) ) and \(B\) has coordinates ( \(q , 2\) ), where \(p\) and \(q\) are integers,

1. find the value of \(p\) and the value of \(q\),
2. find the length of \(A B\), giving your answer as a simplified surd. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\).
3. Find an equation for \(l _ { 2 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.

5. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 3 shows the points \(P , Q\) and \(R\). Points \(P\) and \(Q\) have coordinates ( \(- 1,4\) ) and ( 4,7 ) respectively.

1. Find an equation for the straight line passing through points \(P\) and \(Q\). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers. The point \(R\) has coordinates ( \(p , - 3\) ), where \(p\) is a positive constant. Given that angle \(Q P R = 90 ^ { \circ }\),
2. find the value of \(p\).

1. The line \(l _ { 1 }\) has equation

$$10 x - 2 y + 7 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 1 } { 3 } , \frac { 4 } { 3 } \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

1. The line \(l _ { 1 }\) has equation \(2 x + 3 y = 6\)

The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(( 3 , - 5 )\).
Find the equation for the line \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

3. The line \(l _ { 1 }\) passes through the points \(A ( - 1,4 )\) and \(B ( 5 , - 8 )\)

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is perpendicular to the line \(l _ { 1 }\) and passes through the point \(B ( 5 , - 8 )\)
2. Find an equation for \(l _ { 2 }\) in the form \(a x + b y + c = 0\), where \(a\), b and \(c\) are integers.
   II
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1. The line \(l _ { 1 }\) has equation

$$8 x + 2 y - 15 = 0$$

1. Find the gradient of \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to the line \(l _ { 1 }\) and passes through the point \(\left( - \frac { 3 } { 4 } , 16 \right)\).
2. Find the equation of \(l _ { 2 }\) in the form \(y = m x + c\), where \(m\) and \(c\) are constants.

10. \begin{figure}[h] \end{figure} Diagram NOT drawn to scale The points \(A ( 7 , - 3 ) , B ( 7,20 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 4. The point \(D ( 10,5 )\) is the midpoint of \(A C\).

1. Find the value of \(p\) and the value of \(q\). The line \(l\) passes through \(D\) and is perpendicular to \(A C\).
2. Find an equation for \(l\), in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers. Given that the line \(l\) intersects \(A B\) at \(E\),
3. find the exact coordinates of \(E\).

7. The point \(A\) has coordinates \(( - 1,5 )\) and the point \(B\) has coordinates \(( 4,1 )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find the gradient of \(l\).
2. Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers. The point \(M\) is the midpoint of \(A B\). The point \(C\) has coordinates \(( 5 , k )\) where \(k\) is a constant.
   Given that the distance from \(M\) to \(C\) is \(\sqrt { 13 }\)
3. find the exact possible values of the constant \(k\).
   

12. The circle \(C\) has centre \(A ( 2,1 )\) and passes through the point \(B ( 10,7 )\)

1. Find an equation for \(C\). The line \(l _ { 1 }\) is the tangent to \(C\) at the point \(B\).
2. Find an equation for \(l _ { 1 }\) The line \(l _ { 2 }\) is parallel to \(l _ { 1 }\) and passes through the mid-point of \(A B\).
   Given that \(l _ { 2 }\) intersects \(C\) at the points \(P\) and \(Q\),
3. find the length of \(P Q\), giving your answer in its simplest surd form.

8. \begin{figure}[h] \end{figure} The points \(A ( 1,7 ) , B ( 20,7 )\) and \(C ( p , q )\) form the vertices of a triangle \(A B C\), as shown in Figure 2. The point \(D ( 8,2 )\) is the mid-point of \(A C\).

1. Find the value of \(p\) and the value of \(q\). The line \(l\), which passes through \(D\) and is perpendicular to \(A C\), intersects \(A B\) at \(E\).
2. Find an equation for \(l\), in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.
3. Find the exact \(x\)-coordinate of \(E\).