Question

Find the points on the 𝑦-axis which are at a distance of \[2\sqrt{5}\] units from the point (−4, 7).

Answer 1

Given:

Point: \[(x_1, y_1) = (-4, 7)\]

Distance: \[d = 2\sqrt{5}\text{ units}\]

Since the required point lies on the 𝑦-axis, its abscissa 𝑥-coordinate is 0. Let the required point be \[(x_2, y_2) = (0, y)\].

Formula: \[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

\[2\sqrt{5} = \sqrt{(0 - (-4))^2 + (y - 7)^2}\]

\[2\sqrt{5} = \sqrt{4^2 + (y - 7)^2}\]

\[2\sqrt{5} = \sqrt{16 + (y - 7)^2}\]

Squaring both sides:

\[(2\sqrt{5})^2 = 16 + (y - 7)^2\]

\[20 = 16 + (y - 7)^2\]

\[(y - 7)^2 = 20 - 16\]

\[(y - 7)^2 = 4\]

Taking square root on both sides: \[y - 7 = \pm 2\]

y − 7 = 2

y = 9

y − 7 = −2

y = 5

Hence, the required points on the 𝑦-axis are (0, 5) and (0, 9).