1.03a Straight lines: equation forms y=mx+c, ax+by+c=0

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]

\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 curve \(y = \frac{10}{2x + 1} - 2\) intersects the \(x\)-axis at \(A\). The tangent to the curve at \(A\) intersects the \(y\)-axis at \(C\).

1. Show that the equation of \(AC\) is \(5y + 4x = 8\). [5]
2. Find the distance \(AC\). [2]

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]

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]

\(A(-1, 1)\) and \(P(a, b)\) are two points, where \(a\) and \(b\) are constants. The gradient of \(AP\) is 2.

1. Find an expression for \(b\) in terms of \(a\). [2]
2. \(B(10, -1)\) is a third point such that \(AP = AB\). Calculate the coordinates of the possible positions of \(P\). [6]

\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]

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]

\includegraphics{figure_4} The variables \(x\) and \(y\) satisfy the equation \(ky = e^{cx}\), where \(k\) and \(c\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((2.80, 0.372)\) and \((5.10, 2.21)\), as shown in the diagram. Find the values of \(k\) and \(c\). Give each value correct to 2 significant figures. [4]

\includegraphics{figure_6} The variables \(x\) and \(y\) satisfy the equation \(ay = b^x\), where \(a\) and \(b\) are constants. The graph of \(\ln y\) against \(x\) is a straight line passing through the points \((0.50, 2.24)\) and \((3.40, 8.27)\), as shown in the diagram. Find the values of \(a\) and \(b\). Give each value correct to 1 significant figure. [4]

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]