Gradient from equation or points

1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.

1. Find the length of \(A B\).
2. Find the gradient of the line \(A B\).
3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).

3

1. Find the gradient of the line \(l\) which has equation \(3 x - 5 y - 20 = 0\).
2. The line \(l\) crosses the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Find the coordinates of the mid-point of \(P Q\).

2 The point \(P\) has coordinates \(( \sqrt { 3 } , 2 \sqrt { 3 } )\) and the point \(Q\) has coordinates \(( \sqrt { 5 } , 4 \sqrt { 5 } )\). Show that the gradient of \(P Q\) can be expressed as \(n + \sqrt { 15 }\), stating the value of the integer \(n\).
[0pt] [5 marks]

Line \(L\) has equation $$5y = 4x + 6$$ Find the gradient of a line parallel to line \(L\) Circle your answer. $$\frac{5}{4} \quad -\frac{4}{5} \quad \frac{4}{5} \quad \frac{5}{4}$$ [1 mark]

Find the gradient of the line with equation \(2x + 5y = 7\) Circle your answer. [1 mark] \(\frac{2}{5}\) \quad \(\frac{5}{2}\) \quad \(-\frac{2}{5}\) \quad \(-\frac{5}{2}\)