Key points to solve the problem:
Idea of distance formula- Distance between two points A(x1,y1) and B(x2,y2) is given by- AB = \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
How to approach: To find locus of a point we first assume the coordinate of point to be (h, k) and write a mathematical equation as per the conditions mentioned in question and finally replace (h, k) with (x, y) to get the locus of point.
Let the coordinates of point whose locus is to be determined be (h, k)
As we need to maintain a same distance of (h,k) from (2,4) and y-axis.
So we select point (0,k) on y-axis.
From distance formula:
Distance of (h,k) from (2,4) = \(\sqrt{(h - 2)^2 + (k - 4)^2}\)
Distance of (h,k) from (0,k) = \(\sqrt{(h - 0)^2 + (k - k)^2}\)
According to question both distance are same.
\(\therefore\) \(\sqrt{(h - 2)^2 + (k - 4)^2}\) = \(\sqrt{(h - 0)^2 + (k - k)^2}\)
(h - 2)2 + (k - 4)2 = (h - 0)2 + (k - k)2
\(\Rightarrow\) h2 + 4 - 4h + k2 - 8k + 16 = h2 + 0
\(\Rightarrow\) k2 - 4h - 8k + 20 = 0
Replace (h, k) with (x, y)
Thus, locus of point equidistant from (2,4) and y-axis is
y2 - 4x - 8y + 20 = 0
