Question

The measure of the chord intercepted by circle x²+ y²= 9 and the line x - y + 2 + 0 is ?

1. ​7
2. 5
3. 2 under root 5
4. under root 28

Answer 1

$$\begin{gathered} \mathrm{The}\mathrm{given}\mathrm{equation}\mathrm{of}\mathrm{line}\mathrm{is}: \\ \mathrm{x}-\mathrm{y}+2=0....\left(1\right) \\ \mathrm{The}\mathrm{given}\mathrm{equation}\mathrm{of}\mathrm{circle}\mathrm{is}: \\ {\mathrm{x}}^2+{\mathrm{y}}^2=9.....\left(2\right) \\ \mathrm{The}\mathrm{centre}\mathrm{of}\mathrm{the}\mathrm{circle}\mathrm{is}\mathrm{C}\left(0,0\right)\mathrm{and}\mathrm{radius}3\mathrm{units}. \\ \mathrm{Length}\mathrm{of}\mathrm{the}\perp \mathrm{drawn}\mathrm{from}\mathrm{the}\mathrm{point}\mathrm{C}\mathrm{on}\mathrm{the}\mathrm{line}\mathrm{x}-\mathrm{y}+2=0\mathrm{is}: \\ \mathrm{d}=\frac{\left|0\times 1+0\times \left(-1\right)+2\right|}{\sqrt{{\left(1\right)}^2+{\left(-1\right)}^2}}=\sqrt{2} \\ \mathrm{Since},\mathrm{d}<\hspace{0.17em}\mathrm{r},\mathrm{so}\mathrm{the}\mathrm{line}\mathrm{intersect}\mathrm{the}\mathrm{circle}\mathrm{at}2\mathrm{points}. \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{chord}\mathrm{intercepted}=2\sqrt{{\mathrm{r}}^2-{\mathrm{d}}^2}=2\sqrt{9-2}=2\sqrt{7}=\sqrt{28}\mathrm{units} \end{gathered}$$