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

\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]

\(f(x) = 9 - (x - 2)^2\)

1. Write down the maximum value of \(f(x)\). [1]
2. Sketch the graph of \(y = f(x)\), showing the coordinates of the points at which the graph meets the coordinate axes. [5]

The points \(A\) and \(B\) on the graph of \(y = f(x)\) have coordinates \((-2, -7)\) and \((3, 8)\) respectively.

1. Find, in the form \(y = mx + c\), an equation of the straight line through \(A\) and \(B\). [4]
2. Find the coordinates of the point at which the line \(AB\) crosses the \(x\)-axis. [2]

The mid-point of \(AB\) lies on the line with equation \(y = kx\), where \(k\) is a constant.

1. Find the value of \(k\). [2]

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]

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 \(A\) 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]

\includegraphics{figure_2} Figure 2 shows the curve with equation \(y^2 = 4(x - 2)\) and the line with equation \(2x - 3y = 12\). The curve crosses the \(x\)-axis at the point \(A\), and the line intersects the curve at the points \(P\) and \(Q\).

1. Write down the coordinates of \(A\). [1]
2. Find, using algebra, the coordinates of \(P\) and \(Q\). [6]
3. Show that \(\angle PAQ\) is a right angle. [4]

The points \(A\) and \(B\) have coordinates \((1, 2)\) and \((5, 8)\) respectively.

1. Find the coordinates of the mid-point of \(AB\). [2]
2. Find, in the form \(y = mx + c\), an equation for the straight line through \(A\) and \(B\). [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]

\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]

[In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.] The points \(P(3\cos \alpha, 2\sin \alpha)\) and \(Q(3\cos \beta, 2\sin \beta)\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$

1. Show the equation of the chord \(PQ\) is $$\frac{x}{3}\cos\frac{(\alpha + \beta)}{2} + \frac{y}{2}\sin\frac{(\alpha + \beta)}{2} = \cos\frac{(\alpha - \beta)}{2}$$ [4]
2. Write down the coordinates of the mid-point of \(PQ\). [1]

Given that the gradient, \(m\), of the chord \(PQ\) is a constant,

1. show that the centre of the chord lies on a line $$y = -kx$$ expressing \(k\) in terms of \(m\). [5]

The hyperbola \(C\) has equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)

1. Show that an equation of the normal to \(C\) at \(P(a \sec \theta, b \tan \theta)\) is $$by + ax \sin \theta = (a^2 + b^2)\tan \theta$$ [6] The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(AB\) is \(M\).
2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies. [7]

(Total 13 marks)

\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]

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]

The points \(A(1, 3)\) and \(B(4, 21)\) lie on the curve \(y = x^2 + x + 1\).

1. Find the gradient of the line \(AB\). [2]
2. Find the gradient of the curve \(y = x^2 + x + 1\) at the point where \(x = 3\). [2]

The points \(A\) and \(B\) have coordinates \((4, -2)\) and \((10, 6)\) respectively. \(C\) is the mid-point of \(AB\). Find

1. the coordinates of \(C\), [2]
2. the length of \(AC\), [2]
3. the equation of the circle that has \(AB\) as a diameter, [3]
4. the equation of the tangent to the circle in part (iii) at the point \(A\), giving your answer in the form \(ax + by = c\). [5]

\(A\) 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]

Find, in the form \(y = mx + c\), the equation of the line passing through A\((3, 7)\) and B\((5, -1)\). Show that the midpoint of AB lies on the line \(x + 2y = 10\). [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]