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

The line \(l\) passes through the points \(A(-3, 0)\) and \(B\left(\frac{5}{3}, 22\right)\)

1. Find the equation of \(l\) giving your answer in the form \(y = mx + c\) where \(m\) and \(c\) are constants. [3]

\includegraphics{figure_2} Figure 2 shows the line \(l\) and the curve \(C\), which intersect at \(A\) and \(B\). Given that

• \(C\) has equation \(y = 2x^2 + 5x - 3\)
• the region \(R\), shown shaded in Figure 2, is bounded by \(l\) and \(C\)

1. use inequalities to define \(R\). [2]

A curve has equation \(y = kx^{\frac{1}{2}}\) where \(k\) is a constant. The point \(P\) on the curve has \(x\)-coordinate 4. The normal to the curve at \(P\) is parallel to the line \(2x + 3y = 0\) and meets the \(x\)-axis at the point \(Q\). The line \(PQ\) is the radius of a circle centre \(P\). Show that \(k = \frac{1}{2}\). Find the equation of the circle. [10]

Find the equation of the line passing through the points \((-2, 5)\) and \((4, -7)\). Give your answer in the form \(y = mx + c\). [3]

\includegraphics{figure_2} The diagram shows a triangle \(ABC\). The vertices have coordinates \(A(3, -7)\), \(B(9, 1)\) and \(C(-1, -5)\).

1. 1. Find the length of the side \(AB\). [2]
   2. Find the coordinates of the mid-point of \(AB\). [1]
   3. A circle has diameter \(AB\). Find the equation of the circle in the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\), \(b\) and \(r\) are constants to be found. [3]
2. Find the equation of the line \(l\) passing through \(B\) parallel to \(AC\). [3]