1.03b Straight lines: parallel and perpendicular relationships

The line with gradient \(-2\) passing through the point \(P(3t, 2t)\) intersects the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).

1. Find the area of triangle \(AOB\) in terms of \(t\). [3]

The line through \(P\) perpendicular to \(AB\) intersects the \(x\)-axis at \(C\).

1. Show that the mid-point of \(PC\) lies on the line \(y = x\). [4]

The point \(C\) lies on the perpendicular bisector of the line joining the points \(A(4, 6)\) and \(B(10, 2)\). \(C\) also lies on the line parallel to \(AB\) through \((3, 11)\).

1. Find the equation of the perpendicular bisector of \(AB\). [4]
2. Calculate the coordinates of \(C\). [3]

\includegraphics{figure_9} The diagram shows a trapezium \(ABCD\) in which \(AB\) is parallel to \(DC\) and angle \(BAD\) is \(90°\). The coordinates of \(A\), \(B\) and \(C\) are \((2, 6)\), \((5, -3)\) and \((8, 3)\) respectively.

1. Find the equation of \(AD\). [3]
2. Find, by calculation, the coordinates of \(D\). [3]

The point \(E\) is such that \(ABCE\) is a parallelogram.

1. Find the length of \(BE\). [2]

\(A\) is the point \((a, 2a - 1)\) and \(B\) is the point \((2a + 4, 3a + 9)\), where \(a\) is a constant.

1. Find, in terms of \(a\), the gradient of a line perpendicular to \(AB\). [3]
2. Given that the distance \(AB\) is \(\sqrt{260}\), find the possible values of \(a\). [4]

Three points, \(A\), \(B\) and \(C\), are such that \(B\) is the mid-point of \(AC\). The coordinates of \(A\) are \((2, m)\) and the coordinates of \(B\) are \((n, -6)\), where \(m\) and \(n\) are constants.

1. Find the coordinates of \(C\) in terms of \(m\) and \(n\). [2]

The line \(y = x + 1\) passes through \(C\) and is perpendicular to \(AB\).

1. Find the values of \(m\) and \(n\). [5]

The equation of a curve is \(y = 2x + \frac{12}{x}\) and the equation of a line is \(y + x = k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the line does not meet the curve. [3]

In the case where \(k = 15\), the curve intersects the line at points \(A\) and \(B\).

1. Find the coordinates of \(A\) and \(B\). [3]
2. Find the equation of the perpendicular bisector of the line joining \(A\) and \(B\). [3]

Points \(A\) and \(B\) have coordinates \((h, h)\) and \((4h + 6, 5h)\) respectively. The equation of the perpendicular bisector of \(AB\) is \(3x + 2y = k\). Find the values of the constants \(h\) and \(k\). [7]

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

1. Find and simplify the gradient of \(AB\), showing that it is independent of \(k\). [2]
2. Find and simplify the equation of the perpendicular bisector of \(AB\). [5]

\includegraphics{figure_2} The line \(l_1\) shown in Figure 2 has equation \(2x + 3y = 26\) The line \(l_2\) passes through the origin \(O\) and is perpendicular to \(l_1\)

1. Find an equation for the line \(l_2\) [4]

The line \(l_1\) intersects the line \(l_1\) at the point \(C\). Line \(l_1\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.

1. Find the area of triangle \(OBC\). Give your answer in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers to be found. [6]

\includegraphics{figure_2} The points \(A(1, 7)\), \(B(20, 7)\) and \(C(p, q)\) form the vertices of a triangle \(ABC\), as shown in Figure 2. The point \(D(8, 2)\) is the mid-point of \(AC\).

1. Find the value of \(p\) and the value of \(q\). [2]

The line \(l\), which passes through \(D\) and is perpendicular to \(AC\), intersects \(AB\) at \(E\).

1. Find an equation for \(l\), in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]
2. Find the exact \(x\)-coordinate of \(E\). [2]

The line \(l_1\) passes through the point \((9, -4)\) and has gradient \(\frac{1}{3}\).

1. Find an equation for \(l_1\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

The line \(l_2\) passes through the origin \(O\) and has gradient \(-2\). The lines \(l_1\) and \(l_2\) intersect at the point \(P\).

1. Calculate the coordinates of \(P\). [4]

Given that \(l_1\) crosses the \(y\)-axis at the point \(C\),

1. calculate the exact area of \(\triangle OCP\). [3]

The line \(L\) has equation \(y = 5 - 2x\).

1. Show that the point \(P(3, -1)\) lies on \(L\). [1]
2. Find an equation of the line perpendicular to \(L\), which passes through \(P\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The line \(l_1\) passes through the points \(P(-1, 2)\) and \(Q(11, 8)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\), where \(m\) and \(c\) are constants. [4]

The line \(l_2\) passes through the point \(R(10, 0)\) and is perpendicular to \(l_1\). The lines \(l_1\) and \(l_2\) intersect at the point \(S\).

1. Calculate the coordinates of \(S\). [5]
2. Show that the length of \(RS\) is \(3\sqrt{5}\). [2]
3. Hence, or otherwise, find the exact area of triangle \(PQR\). [4]

\includegraphics{figure_2} The points \(A(3, 0)\) and \(B(0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.

1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]

The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.

1. Find the length of \(BC\) in terms of \(k\). [2]

Given that the length of \(BC\) is 10 and using your answer to part (b),

1. find the value of \(k\), [4]
2. find the coordinates of \(D\). [2]

The points \(A(-1, -2)\), \(B(7, 2)\) and \(C(k, 4)\), where \(k\) is a constant, are the vertices of \(\triangle ABC\). Angle \(ABC\) is a right angle.

1. Find the gradient of \(AB\). [2]
2. Calculate the value of \(k\). [2]
3. Show that the length of \(AB\) may be written in the form \(p\sqrt{5}\), where \(p\) is an integer to be found. [3]
4. Find the exact value of the area of \(\triangle ABC\). [3]
5. Find an equation for the straight line \(l\) passing through \(B\) and \(C\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [2]

The straight line \(l_1\) has equation \(4y + x = 0\). The straight line \(l_2\) has equation \(y = 2x - 3\).

1. On the same axes, sketch the graphs of \(l_1\) and \(l_2\). Show clearly the coordinates of all points at which the graphs meet the coordinate axes. [3]

The lines \(l_1\) and \(l_2\) intersect at the point \(A\).

1. Calculate, as exact fractions, the coordinates of \(A\). [3]
2. Find an equation of the line through \(A\) which is perpendicular to \(l_1\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [3]

\includegraphics{figure_3} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 3.

1. Find the coordinates of the centre of the circle. [2]
2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
3. Find an equation of tangent to the circle at \(B\). [3]

The line \(l\) passes through \(A\) and the origin.

1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]

The straight line \(l_1\) with equation \(y = \frac{3}{2}x - 2\) crosses the \(y\)-axis at the point \(P\). The point \(Q\) has coordinates \((5, -3)\). The straight line \(l_2\) is perpendicular to \(l_1\) and passes through \(Q\).

1. Calculate the coordinates of the mid-point of \(PQ\). [3]
2. Find an equation for \(l_2\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integer constants. [4]

The lines \(l_1\) and \(l_2\) intersect at the point \(R\).

1. Calculate the exact coordinates of \(R\). [4]

\includegraphics{figure_1} The points \(A\) and \(B\) have coordinates \((2, -3)\) and \((8, 5)\) respectively, and \(AB\) is a chord of a circle with centre \(C\), as shown in Fig. 1.

1. Find the gradient of \(AB\). [2]

The point \(M\) is the mid-point of \(AB\).

1. Find an equation for the line through \(C\) and \(M\). [5]

Given that the \(x\)-coordinate of \(C\) is 4,

1. find the \(y\)-coordinate of \(C\), [2]
2. show that the radius of the circle is \(\frac{5\sqrt{17}}{4}\). [4]

The points \(A\) and \(B\) have coordinates \((3, 4)\) and \((7, -6)\) respectively. The straight line \(l\) passes through \(A\) and is perpendicular to \(AB\). Find an equation for \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [5]

The circle \(C\), with centre at the point \(A\), has equation \(x^2 + y^2 - 10x + 9 = 0\). Find

1. the coordinates of \(A\), [2]
2. the radius of \(C\), [2]
3. the coordinates of the points at which \(C\) crosses the \(x\)-axis. [2]

Given that the line \(l\) with gradient \(\frac{7}{T}\) is a tangent to \(C\), and that \(l\) touches \(C\) at the point \(T\),

1. find an equation of the line which passes through \(A\) and \(T\). [3]

\includegraphics{figure_3} The points \(A\) and \(B\) lie on a circle with centre \(P\), as shown in Figure 3. The point \(A\) has coordinates \((1, -2)\) and the mid-point \(M\) of \(AB\) has coordinates \((3, 1)\). The line \(l\) passes through the points \(M\) and \(P\).

1. Find an equation for \(l\). [4]

Given that the \(x\)-coordinate of \(P\) is 6,

1. use your answer to part (a) to show that the \(y\)-coordinate of \(P\) is \(-1\). [1]
2. find an equation for the circle. [4]

\includegraphics{figure_2} The straight line \(l_1\) has equation \(2y = 3x + 7\) The line \(l_1\) crosses the \(y\)-axis at the point \(A\) as shown in Figure 2.

1. 1. State the gradient of \(l_1\)
   2. Write down the coordinates of the point \(A\). [2]

Another straight line \(l_2\) intersects \(l_1\) at the point \(B(1, 5)\) and crosses the \(x\)-axis at the point \(C\), as shown in Figure 2. Given that \(\angle ABC = 90°\),

1. find an equation of \(l_2\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The rectangle \(ABCD\), shown shaded in Figure 2, has vertices at the points \(A\), \(B\), \(C\) and \(D\).

1. Find the exact area of rectangle \(ABCD\). [5]

\includegraphics{figure_1} The points \(A(-3, -2)\) and \(B(8, 4)\) are at the ends of a diameter of the circle shown in Fig. 1.

1. Find the coordinates of the centre of the circle. [2]
2. Find an equation of the diameter \(AB\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]
3. Find an equation of tangent to the circle at \(B\). [3]

The line \(l\) passes through \(A\) and the origin.

1. Find the coordinates of the point at which \(l\) intersects the tangent to the circle at \(B\), giving your answer as exact fractions. [4]

\includegraphics{figure_2} The points \(A (3, 0)\) and \(B (0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 2.

1. Write down the gradient of \(AB\) and hence the gradient of \(BC\). [3]

The point \(C\) has coordinates \((8, k)\), where \(k\) is a positive constant.

1. Find the length of \(BC\) in terms of \(k\). [2]

Given that the length of \(BC\) is 10 and using your answer to part (b),

1. find the value of \(k\), [4]
2. find the coordinates of \(D\). [2]