Question

The latus rectum of the parabola whose directrix is x + y – 2 = 0 and the focus is (3, –4), is __________.

Answer 1

For a given parabola
vertex is given by (0, 0), focus is (3, 4) and directrix is x + y − 2 = 0
Distance between directrix and focus is 2a

$$\begin{gathered} \mathrm{i}.\mathrm{e}.\left|\frac{3-4-2}{\sqrt{1+1}}\right|=2a \\ \mathrm{i}.\mathrm{e}.\frac{3}{\sqrt{2}}=2a \\ \mathrm{i}.\mathrm{e}.a=\frac{3}{2\sqrt{2}} \end{gathered}$$
$\begin{array}{rcl}\therefore \mathrm{Length}\mathrm{of}\mathrm{latus}\mathrm{rectum}\mathrm{is}4a& =& 4\times \frac{3}{2\sqrt{2}}=\frac{2\times 3}{\sqrt{2}}=3\sqrt{2}\\ & =& 3\sqrt{2}\end{array}$