Question

The number of points on x-axis which are at a distance of $2\sqrt{5}$ from the point (7, –4), is __________.

Answer 1

The distance between $\left(x_1,y_1\right)\mathrm{and}\left(x_2,y_2\right)\mathrm{is}\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

Let the point on x-axis be P(x, 0) which is at a distance of $2\sqrt{5}$ from the point (7, –4).

Therefore, the distance between the points (x, 0) and (7, –4)$=\sqrt{{\left(7-x\right)}^2+{\left(-4-0\right)}^2}$
$$\begin{gathered} \Rightarrow 2\sqrt{5}=\sqrt{{\left(7-x\right)}^2+{\left(-4\right)}^2} \\ \mathrm{Squaring}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get} \\ \Rightarrow {\left(2\sqrt{5}\right)}^2={\left(\sqrt{{\left(7-x\right)}^2+16}\right)}^2 \\ \Rightarrow 20={\left(7-x\right)}^2+16 \\ \Rightarrow 20=49+x^2-14x+16 \\ \Rightarrow x^2-14x+65-20=0 \\ \Rightarrow x^2-14x+45=0 \\ \Rightarrow x^2-9x-5x+45=0 \\ \Rightarrow x\left(x-9\right)-5\left(x-9\right)=0 \\ \Rightarrow \left(x-5\right)\left(x-9\right)=0 \\ \Rightarrow x=5,9 \end{gathered}$$

Hence, the number of points on x-axis which are at a distance of $2\sqrt{5}$ from the point (7, –4), is 2.