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
= \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
Idea of section formula- Let two points A(x1,y1) and B(x2,y2) forms a line segment. If a point C(x, y) divides line segment AB in ratio of m:n internally, then coordinates of C is given as:
C = \(\bigg(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\bigg)\)when m = n = 1 , C becomes the midpoint of AB and C is given as C = \(\bigg(\frac{x_2 + x_1}{2} , \frac{y_2 + y_1}{2}\bigg)\)
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). Name the moving point be C
Given that (h,k) is midpoint of line x cos α + y sin α = p intercepted between axes.
So we need to first find the points at which x cos α + y sin α = p
cuts the axes after which we will apply the section formula to get the locus.
Put y = 0
∴ x = p/cos α ⇒ coordinates on x-axis is (p/cos α , 0).
Name the point A Similarly, Put x = 0
∴ y = p/sin α ⇒ coordinates on y-axis is (0, p/sin α ).
Name this point B
As C(h,k) is midpoint of AB
∴ coordinate of C is given by:
Replace (h, k) with (x, y)
Thus, locus of point on rod is: \(\frac{p^2}{4x^2} + \frac{p^2}{4y^2} = 1\)
