1.03b Straight lines: parallel and perpendicular relationships

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_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 \((4, 6)\) and \((12, 2)\) respectively. The straight line \(l_1\) passes through \(A\) and \(B\).

1. Find an equation for \(l_1\) in the form \(ax + by = c\), where \(a\), \(b\) and \(c\) are integers. [4]

The straight line \(l_2\) passes through the origin and has gradient \(-4\).

1. Write down an equation for \(l_2\). [1]

The lines \(l_1\) and \(l_2\) intercept at the point \(C\).

1. Find the exact coordinates of the mid-point of \(AC\). [5]

\includegraphics{figure_1} The points \(A(3, 0)\) and \(B(0, 4)\) are two vertices of the rectangle \(ABCD\), as shown in Fig. 1.

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]

\(A\) is the point \((-2, 6)\) and \(B\) is the point \((3, -8)\). The line \(l\) is perpendicular to the line \(x - 3y + 15 = 0\) and passes through the mid-point of \(AB\). Find the equation of \(l\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [7]

\(A\) is the point \((5, 7)\) and \(B\) is the point \((-1, -5)\).

1. Find the coordinates of the mid-point of the line segment \(AB\). [2]
2. Find an equation of the line through \(A\) that is perpendicular to the line segment \(AB\), giving your answer in the form \(ax + by + c = 0\) where \(a\), \(b\) and \(c\) are integers. [5]

The line \(L\) is parallel to \(y = -2x + 1\) and passes through the point \((5, 2)\). Find the coordinates of the points of intersection of \(L\) with the axes. [5]

\includegraphics{figure_7} The line AB has equation \(y = 4x - 5\) and passes through the point B(2, 3), as shown in Fig. 7. The line BC is perpendicular to AB and cuts the \(x\)-axis at C. Find the equation of the line BC and the \(x\)-coordinate of C. [5]

A line has equation \(3x + 2y = 6\). Find the equation of the line parallel to this which passes through the point \((2, 10)\). [3]

A\((9, 8)\), B\((5, 0)\) and C\((3, 1)\) are three points.

1. Show that AB and BC are perpendicular. [3]
2. Find the equation of the circle with AC as diameter. You need not simplify your answer. Show that B lies on this circle. [6]
3. BD is a diameter of the circle. Find the coordinates of D. [3]

\includegraphics{figure_11} Fig. 11 shows the line joining the points A \((0, 3)\) and B \((6, 1)\).

1. Find the equation of the line perpendicular to AB that passes through the origin, O. [2]
2. Find the coordinates of the point where this perpendicular meets AB. [4]
3. Show that the perpendicular distance of AB from the origin is \(\frac{9\sqrt{10}}{10}\). [2]
4. Find the length of AB, expressing your answer in the form \(a\sqrt{10}\). [2]
5. Find the area of triangle OAB. [2]

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]
2. State, with a reason, whether or not this circle intersects the \(y\)-axis. [2]
3. Find the equation of the line parallel to the line \(y = 2x\) that passes through the centre of the circle. [2]
4. Show that the line \(y = 2x + 2\) is a tangent to the circle. State the coordinates of the point of contact. [5]

Find the equation of the line which is parallel to \(y = 3x + 1\) and which passes through the point with coordinates \((4, 5)\). [3]

\includegraphics{figure_11} Fig. 11 shows the line through the points A \((-1, 3)\) and B \((5, 1)\).

1. Find the equation of the line through A and B. [3]
2. Show that the area of the triangle bounded by the axes and the line through A and B is \(\frac{32}{3}\) square units. [2]
3. Show that the equation of the perpendicular bisector of AB is \(y = 3x - 4\). [3]
4. A circle passing through A and B has its centre on the line \(x = 3\). Find the centre of the circle and hence find the radius and equation of the circle. [4]

A line \(L\) is parallel to the line \(x + 2y = 6\) and passes through the point \((10, 1)\). Find the area of the region bounded by the line \(L\) and the axes. [5]

Find the equation of the line with gradient \(-2\) which passes through the point \((3, 1)\). Give your answer in the form \(y = ax + b\). Find also the points of intersection of this line with the axes. [3]

\includegraphics{figure_10} Fig. 10 is a sketch of quadrilateral ABCD with vertices A \((1, 5)\), B \((-1, 1)\), C \((3, -1)\) and D \((11, 5)\).

1. Show that \(AB = BC\). [3]
2. Show that the diagonals AC and BD are perpendicular. [3]
3. Find the midpoint of AC. Show that BD bisects AC but AC does not bisect BD. [5]

Find the equation of the line which is perpendicular to the line \(y = 2x - 5\) and which passes through the point \((4, 1)\). Give your answer in the form \(y = ax + b\). [3]

The points \(P\) and \(Q\) have coordinates \((7, 4)\) and \((9, 7)\) respectively.

1. Find an equation for the straight line \(l\) which passes through \(P\) and \(Q\). Give your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The straight line \(m\) has gradient \(8\) and passes through the origin, \(O\).

1. Write down an equation for \(m\). [1]

The lines \(l\) and \(m\) intersect at the point \(R\).

1. Show that \(OP = OR\). [5]

The straight line \(l_1\) has equation \(2x + y - 14 = 0\) and crosses the \(x\)-axis at the point \(A\).

1. Find the coordinates of \(A\). [2]

The straight line \(l_2\) is parallel to \(l_1\) and passes through the point \(B(-6, 6)\).

1. Find an equation for \(l_2\) in the form \(y = mx + c\). [3]

The line \(l_2\) crosses the \(x\)-axis at the point \(C\).

1. Find the coordinates of \(C\). [1]

The point \(D\) lies on \(l_1\) and is such that \(CD\) is perpendicular to \(l_1\).

1. Show that \(D\) has coordinates \((5, 4)\). [5]
2. Find the area of triangle \(ACD\). [2]

The straight line \(l\) has the equation \(x - 5y = 7\). The straight line \(m\) is perpendicular to \(l\) and passes through the point \((-4, 1)\). Find an equation for \(m\) in the form \(y = mx + c\). [5]

The straight line \(l_1\) has gradient \(\frac{3}{4}\) and passes through the point \(A(5, 3)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]

The straight line \(l_2\) has the equation \(3x - 4y + 3 = 0\) and intersects \(l_1\) at the point \(B\).

1. Find the coordinates of \(B\). [3]
2. Find the coordinates of the mid-point of \(AB\). [2]
3. Show that the straight line parallel to \(l_2\) which passes through the mid-point of \(AB\) also passes through the origin. [4]

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

1. Find an equation for \(l\) in the form \(y = mx + c\). [2]

The straight line \(m\) has the equation \(x - 7y + 14 = 0\). Given that \(m\) crosses the \(y\)-axis at the point \(B\) and intersects \(l\) at the point \(C\),

1. find the coordinates of \(B\) and \(C\), [4]
2. show that \(\angle BAC = 90°\), [4]
3. find the area of triangle \(ABC\). [4]

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 \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The straight line \(m\) has the equation \(2x + ky + 7 = 0\), where \(k\) is a constant. Given that \(l\) and \(m\) are perpendicular,

1. find the value of \(k\). [4]

The straight line \(l_1\) has gradient 2 and passes through the point with coordinates \((4, -5)\).

1. Find an equation for \(l_1\) in the form \(y = mx + c\). [2]

The straight line \(l_2\) is perpendicular to the line with equation \(3x - y = 4\) and passes through the point with coordinates \((3, 0)\).

1. Find an equation for \(l_2\). [3]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [3]