1.03b Straight lines: parallel and perpendicular relationships

2 The line \(A B\) has equation \(3 x + 5 y = 8\) and the point \(A\) has coordinates (6, -2).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the straight line which is perpendicular to \(A B\) and which passes through \(A\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 3\) at the point \(B\). Find the coordinates of \(B\).
3. The point \(C\) has coordinates \(( 2 , k )\) and the distance from \(A\) to \(C\) is 5 . Find the two possible values of the constant \(k\).

1 The triangle \(A B C\) has vertices \(A ( - 2,3 ) , B ( 4,1 )\) and \(C ( 2 , - 5 )\).

1. Find the coordinates of the mid-point of \(B C\).
2. 1. Find the gradient of \(A B\), in its simplest form.
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(x + q y = r\), where \(q\) and \(r\) are integers.
   3. Find an equation of the line passing through \(C\) which is parallel to \(A B\).
3. Prove that angle \(A B C\) is a right angle.

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.

6

1. The polynomial \(\mathrm { p } ( x )\) is given by \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\).
   1. Use the Factor Theorem to show that \(x + 1\) is a factor of \(\mathrm { p } ( x )\).
   2. Express \(\mathrm { p } ( x ) = x ^ { 3 } - 7 x - 6\) as the product of three linear factors.
2. The curve with equation \(y = x ^ { 3 } - 7 x - 6\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{de4f827d-f237-488a-9177-3d85d0cb1771-4_403_762_651_641} The curve cuts the \(x\)-axis at the point \(A\) and the points \(B ( - 1,0 )\) and \(C ( 3,0 )\).
   1. State the coordinates of the point \(A\).
   2. Find \(\int _ { - 1 } ^ { 3 } \left( x ^ { 3 } - 7 x - 6 \right) \mathrm { d } x\).
   3. Hence find the area of the shaded region bounded by the curve \(y = x ^ { 3 } - 7 x - 6\) and the \(x\)-axis between \(B\) and \(C\).
   4. Find the gradient of the curve \(y = x ^ { 3 } - 7 x - 6\) at the point \(B\).
   5. Hence find an equation of the normal to the curve at the point \(B\).

1 The points \(A\) and \(B\) have coordinates \(( 6 , - 1 )\) and \(( 2,5 )\) respectively.

1. 1. Show that the gradient of \(A B\) is \(- \frac { 3 } { 2 }\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
2. 1. Find an equation of the line which passes through \(B\) and which is perpendicular to the line \(A B\).
   2. The point \(C\) has coordinates ( \(k , 7\) ) and angle \(A B C\) is a right angle. Find the value of the constant \(k\).

1 The line \(A B\) has equation \(3 x + 5 y = 11\).

1. 1. Find the gradient of \(A B\).
   2. The point \(A\) has coordinates (2,1). Find an equation of the line which passes through the point \(A\) and which is perpendicular to \(A B\).
2. The line \(A B\) intersects the line with equation \(2 x + 3 y = 8\) at the point \(C\). Find the coordinates of \(C\).

5

1. Find the equation of the line perpendicular to \(L\) which passes through \(P\). 5 The line \(L\) has equation 5
2. Hence, find the shortest distance from \(P\) to \(L\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-05_2488_1716_219_153}

8 The lines \(L _ { 1 }\) and \(L _ { 2 }\) are parallel. \(L _ { 1 }\) has equation $$5 x + 3 y = 15$$ and \(L _ { 2 }\) has equation $$5 x + 3 y = 83$$ \(L _ { 1 }\) intersects the \(y\)-axis at the point \(P\).
The point \(Q\) is the point on \(L _ { 2 }\) closest to \(P\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561} 8

1. 1. Find the coordinates of \(Q\).
      8
      1. (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found. 8
   2. A circle, \(C\), has centre ( \(a , - 17\) ). \(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
      8
   3. 1. Find \(a\).
         8
   4. (ii) Find the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}

9 The points \(P\) and \(Q\) have coordinates ( \(- 6,15\) ) and (12, 19) respectively. 9

1. 1. Find the coordinates of the midpoint of \(P Q\) 9
      1. (ii) Find the equation of the perpendicular bisector of \(P Q\) Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
         [0pt] [4 marks]
         9
   2. 1. A circle passes through the points \(P\) and \(Q\) The centre of the circle lies on the line with equation \(2 x - 5 y = - 30\) Find the equation of the circle. 9
   3. (ii) The circle intersects the coordinate axes at \(n\) points.
      State the value of \(n\) $$y = \sin x ^ { \circ }$$ for \(- 360 \leq x \leq 360\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-12_613_1552_532_246}

The line \(l _ { 1 }\) passes through the point \(A ( - 5,20 )\) and the point \(B ( 3 , - 4 )\).

1. Find an equation for \(l _ { 1 }\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants. The line \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\) and passes through the midpoint of \(A B\)
2. Find an equation for \(l _ { 2 }\) giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.

2 Find the equation of the line perpendicular to the line \(y = 5 x\) which passes through the point \(( 2,11 )\). Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers to be found.

5 A circle has equation \(x ^ { 2 } - 6 x + y ^ { 2 } - 4 y = 12\).

1. Show that the centre of the circle is at the point \(( 3,2 )\) and find the radius.
2. \(P Q\) is a diameter of the circle where \(P\) has coordinates \(( - 1 , - 1 )\). Find the equation of \(P Q\), giving your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
3. Another diameter of the circle passes through the point \(( 0,6 )\). Show that this diameter is perpendicular to \(P Q\).

1 Find the equation of the line perpendicular to the line \(y = 5 x + 6\) which passes through the point \(( 1,11 )\). Give your answer in the form \(y = m x + c\).

3

1. The points \(A\) and \(B\) have coordinates ( \(- 4,4\) ) and ( 8,1 ) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).

3

1. The points \(A\) and \(B\) have coordinates \(( - 4,4 )\) and \(( 8,1 )\) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).

The points \(A\) and \(B\) have coordinates \(( - 1,10 )\) and \(( 5,1 )\) respectively. The straight line \(L\) has equation \(2 x - 3 y + 6 = 0\). a) The line \(L\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
b) Determine the ratio in which the line \(L\) divides the line \(A B\).
c) The line \(L\) crosses the \(x\)-axis at the point \(D\). Find the coordinates of \(D\).
d) i) Show that \(L\) is perpendicular to \(A B\).
ii) Calculate the area of the triangle \(A C D\).

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. [5]
2. For \(a = 4\), find the equation of the normal to the circle at \(P\). [4]
3. For \(a = 4\), find the equations of the two tangents to the circle which are parallel to the normal found in (b). [4]

The equation of a circle is \((x - 3)^2 + y^2 = 18\). The line with equation \(y = mx + c\) passes through the point \((0, -9)\) and is a tangent to the circle. Find the two possible values of \(m\) and, for each value of \(m\), find the coordinates of the point at which the tangent touches the circle. [8]

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

1. Find the two possible values of the constant \(a\). [5]
2. For the greater value of \(a\), find the equation of the diameter which is perpendicular to the given tangent. [3]

The coordinates of points \(A\), \(B\) and \(C\) are \((6, 4)\), \((p, 7)\) and \((14, 18)\) respectively, where \(p\) is a constant. The line \(AB\) is perpendicular to the line \(BC\).

1. Given that \(p < 10\), find the value of \(p\). [4]

A circle passes through the points \(A\), \(B\) and \(C\).

1. Find the equation of the circle. [3]
2. Find the equation of the tangent to the circle at \(C\), giving the answer in the form \(dx + ey + f = 0\), where \(d\), \(e\) and \(f\) are integers. [3]

The equation of a circle is \(x^2 + y^2 + px + 2y + q = 0\), 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 \(q\). [2]

The line with equation \(x + 2y = 10\) is the tangent to the circle at the point \(A(4, 3)\).

1. 1. Find the equation of the normal to the circle at the point \(A\). [3]
   2. Find the values of \(p\) and \(q\). [5]

\includegraphics{figure_4} In the diagram, \(A\) is the point \((-1, 3)\) and \(B\) is the point \((3, 1)\). The line \(L_1\) passes through \(A\) and is parallel to \(OB\). The line \(L_2\) passes through \(B\) and is perpendicular to \(AB\). The lines \(L_1\) and \(L_2\) meet at \(C\). Find the coordinates of \(C\). [6]

The line \(L_1\) passes through the points \(A(2, 5)\) and \(B(10, 9)\). The line \(L_2\) is parallel to \(L_1\) and passes through the origin. The point \(C\) lies on \(L_2\) such that \(AC\) is perpendicular to \(L_2\). Find

1. the coordinates of \(C\), [5]
2. the distance \(AC\). [2]

The point \(A\) has coordinates \((-1, -5)\) and the point \(B\) has coordinates \((7, 1)\). The perpendicular bisector of \(AB\) meets the \(x\)-axis at \(C\) and the \(y\)-axis at \(D\). Calculate the length of \(CD\). [6]

The equation of a line is \(2y + x = k\), where \(k\) is a constant, and the equation of a curve is \(xy = 6\).

1. In the case where \(k = 8\), the line intersects the curve at the points \(A\) and \(B\). Find the equation of the perpendicular bisector of the line \(AB\). [6]
2. Find the set of values of \(k\) for which the line \(2y + x = k\) intersects the curve \(xy = 6\) at two distinct points. [3]