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# Find the locus of the mid-point of the portion of the x cos α + y sin α = p which is intercepted between the axes.

Answer :

**Key points to solve the problem:**

• **Idea of distance formula-** Distance between two points A(x_{1},y_{1}) and B(x_{2},y_{2}) is given by- **AB =**

• **Idea of section formula-** Let two points A(x_{1},y_{1}) and B(x_{2},y_{2}) forms a line segment. If a point C(x,y) divides line segment AB in the ratio of m:n internally, then coordinates of C is given as:

**C =** when m = n =1 , C becomes the midpoint of AB and C is given as **C =**

**How to approach:** To find the locus of a point we first assume the coordinate of the point to be (h, k) and write a mathematical equation as per the conditions mentioned in the question and finally replace (h, k) with (x, y) to get the locus of the point.

Let the coordinates of a point whose locus is to be determined to be (h, k). Name the moving point to be C

Given that (h,k) is the midpoint of line x cos α + y sin α = p intercepted between axes.

So we need to 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 the midpoint of AB

∴ coordinate of C is given by:

**C =**

Thus,

**…equation 1**

and **…equation 2**

Squaring and adding equation 1 and 2:

⇒

Replace (h,k) with (x,y)

Thus, the locus of a point on the rod is: **….ans**

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