1.03b Straight lines: parallel and perpendicular relationships

7. The rectangular hyperbola, \(H\), has cartesian equation \(x y = 25\) The point \(P \left( 5 p , \frac { 5 } { p } \right)\), and the point \(Q \left( 5 q , \frac { 5 } { q } \right)\), where \(p , q \neq 0 , p \neq q\), are points on the rectangular hyperbola \(H\).

1. Show that the equation of the tangent at point \(P\) is $$p ^ { 2 } y + x = 10 p$$
2. Write down the equation of the tangent at point \(Q\). The tangents at \(P\) and \(Q\) meet at the point \(N\).
   Given \(p + q \neq 0\),
3. show that point \(N\) has coordinates \(\left( \frac { 10 p q } { p + q } , \frac { 10 } { p + q } \right)\). The line joining \(N\) to the origin is perpendicular to the line \(P Q\).
4. Find the value of \(p ^ { 2 } q ^ { 2 }\).

3. A rectangular hyperbola has parametric equations $$x = 2 t , \quad y = \frac { 2 } { t } , \quad t \neq 0$$ Points \(P\) and \(Q\) on this hyperbola have parameters \(t = \frac { 1 } { 2 }\) and \(t = 4\) respectively.
The line \(L\), which passes through the origin \(O\), is perpendicular to the chord \(P Q\).
Find an equation for \(L\).

The line \(l _ { 1 }\) has equation \(3 x + 5 y - 2 = 0\)

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

The points \(P\) and \(Q\) have coordinates \(( - 1,6 )\) and \(( 9,0 )\) respectively. The line \(l\) is perpendicular to \(P Q\) and passes through the mid-point of \(P Q\).
Find an equation for \(l\), giving your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers.

10 The points \(D , E\) and \(F\) have coordinates \(( - 2,0 ) , ( 0 , - 1 )\) and \(( 2,3 )\) respectively.

1. Calculate the gradient of \(D E\).
2. Find the equation of the line through \(F\), parallel to \(D E\), giving your answer in the form \(a x + b y + c = 0\).
3. By calculating the gradient of \(E F\), show that \(D E F\) is a right-angled triangle.
4. Calculate the length of \(D F\).
5. Use the results of parts (iii) and (iv) to show that the circle which passes through \(D , E\) and \(F\) has equation \(x ^ { 2 } + y ^ { 2 } - 3 y - 4 = 0\).

7

1. Find the gradient of the line \(l\) which has equation \(x + 2 y = 4\).
2. Find the equation of the line parallel to \(l\) which passes through the point ( 6,5 ), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
3. Solve the simultaneous equations $$y = x ^ { 2 } + x + 1 \quad \text { and } \quad x + 2 y = 4$$

9

1. Find the gradient of the line \(l _ { 1 }\) which has equation \(4 x - 3 y + 5 = 0\).
2. Find an equation of the line \(l _ { 2 }\), which passes through the point ( 1,2 ) and which is perpendicular to the line \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\). The line \(l _ { 1 }\) crosses the \(x\)-axis at \(P\) and the line \(l _ { 2 }\) crosses the \(y\)-axis at \(Q\).
3. Find the coordinates of the mid-point of \(P Q\).
4. Calculate the length of \(P Q\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.

5

1. Find the gradient of the line \(4 x + 5 y = 24\).
2. A line parallel to \(4 x + 5 y = 24\) passes through the point \(( 0,12 )\). Find the coordinates of its point of intersection with the \(x\)-axis.

12

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.

2 A line \(L\) is parallel to \(y = 4 x + 5\) and passes through the point \(( - 1,6 )\). Find the equation of the line \(L\) in the form \(y = a x + b\). Find also the coordinates of its intersections with the axes.

10 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

10 \begin{figure}[h] \end{figure} In Fig.10, A has coordinates \(( 1,1 )\) and C has coordinates \(( 3,5 )\). M is the mid-point of AC . The line \(l\) is perpendicular to AC.

1. Find the coordinates of M . Hence find the equation of \(l\).
2. The point B has coordinates \(( - 2,5 )\). Show that B lies on the line \(l\).
   Find the coordinates of the point D such that ABCD is a rhombus.
3. Find the lengths MC and MB . Hence calculate the area of the rhombus ABCD .

4 The coordinates of the points \(\mathrm { A } , \mathrm { B }\) and C are ( \(- 2,2\) ), ( 1,3 ) and ( \(3 , - 3\) ) respectively.

1. Find the gradients of the lines AB and BC .
2. Show that the triangle ABC is a right-angled triangle.
3. Find the area of the triangle ABC .

8 Find the equation of the line that passes through the point \(( 1,2 )\) and is perpendicular to the line \(3 x + 2 y = 5\).

5 The vertices of a triangle have coordinates ( 1,5 ), ( \(- 3,7\) ) and ( \(- 2 , - 1\) ).
Show that the triangle is right-angled.

12 ABCD is a parallelogram. The coordinates of \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D are (-2, 3), (2, 4), (8, -3) and ( \(4 , - 4\) ) respectively. \includegraphics[max width=\textwidth, alt={}, center]{3b6291ef-bef9-49de-a20f-591e621bed65-4_592_725_387_492}

1. Prove that AB and BD are perpendicular.
2. Find the lengths of AB and BD and hence find the area of the parallelogram ABCD
3. Find the equation of the line CD and show that it meets the \(y\)-axis at \(\mathrm { X } ( 0 , - 5 )\).
4. Show that the lines BX and AD bisect each other.
5. Explain why the area of the parallelogram ABCD is equal to the area of the triangle BXC.
   Find the length of BX and hence calculate exactly the perpendicular distance of C from BX .

2 Find the equation of the straight line which is parallel to the line \(y = 3 x + 5\) and which goes through the point \(( 2,12 )\).

13 In Fig.13, XP and XQ are the perpendicular bisectors of AC and BC respectively. \begin{figure}[h] \end{figure}

1. Find the coordinates of X .
2. Hence show that \(\mathrm { AX } = \mathrm { BX } = \mathrm { CX }\).
3. The circumcircle of a triangle is the circle which passes through the vertices of the triangle.
   Write down the equation of the circumcircle of the triangle ABC .
4. Find the coordinates of the points where the circle cuts the \(x\) axis.

10. The straight line \(l\) has gradient 3 and passes through the point \(A ( - 6,4 )\).

1. Find an equation for \(l\) in the form \(y = m x + c\). The straight line \(m\) has the equation \(x - 7 y + 14 = 0\).
   Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),
2. find the coordinates of \(B\) and \(C\),
3. show that \(\angle B A C = 90 ^ { \circ }\),
4. find the area of triangle \(A B C\).

2. The points \(A , B\) and \(C\) have coordinates \(( - 3,0 ) , ( 5 , - 2 )\) and \(( 4,1 )\) respectively. Find an equation for the straight line which passes through \(C\) and is parallel to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.

1. The straight line \(l\) has the equation \(x - 5 y = 7\).

The straight line \(m\) is perpendicular to \(l\) and passes through the point \(( - 4,1 )\).
Find an equation for \(m\) in the form \(y = m x + c\).

9. The straight line \(l _ { 1 }\) passes through the point \(A ( - 2,5 )\) and the point \(B ( 4,1 )\).

1. Find an equation for \(l _ { 1 }\) in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers. The straight line \(l _ { 2 }\) passes through \(B\) and is perpendicular to \(l _ { 1 }\).
2. Find an equation for \(l _ { 2 }\). Given that \(l _ { 2 }\) meets the \(y\)-axis at the point \(C\),
3. show that triangle \(A B C\) is isosceles.

7. The straight line \(l\) passes through the point \(P ( - 3,6 )\) and the point \(Q ( 1 , - 4 )\).

1. Find an equation for \(l\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The straight line \(m\) has the equation \(2 x + k y + 7 = 0\), where \(k\) is a constant.
   Given that \(l\) and \(m\) are perpendicular,
2. find the value of \(k\).

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.

3 \begin{figure}[h] \end{figure} Not to scale A circle has centre \(\mathrm { C } ( 1,3 )\) and passes through the point \(\mathrm { A } ( 3,7 )\) 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.