it's easy , find the equation of chord of contact ,t=o then assume points to be (h,k) and equate line distant from this point to (3,0)

Introduction:
We are given that tangents are drawn from point P(3,0) to a circle with center at the origin, point A, and point B. We are asked to find the equation of the locus of the point whose distances from point P and the line AB are equal.

Understanding the problem:
To solve this problem, we need to find the equation of the locus, which represents all the points that satisfy the given condition. The locus will be a curve that passes through all the points whose distances from point P and the line AB are equal.

Steps to find the equation of the locus:

Step 1: Determine the coordinates of point A and point B:
Since point A is the center of the circle and the origin, its coordinates are (0,0). To find the coordinates of point B, we need to find the equation of the line AB.

Step 2: Find the equation of the line AB:
Since point P(3,0) lies on the tangent, the slope of the line AB can be found using the formula: m = -1/slope of tangent. The slope of the tangent can be found by considering the slope of the line joining the center of the circle (0,0) and the point of tangency (3,0). Therefore, the slope of the tangent is 0/3 = 0. Thus, the slope of the line AB is -1/0, which is undefined. This means the line AB is a vertical line passing through the point A(0,0).

Step 3: Find the equation of the locus:
Since the line AB is vertical and passes through the point A(0,0), any point on this line can be represented by the coordinates (0,y). Let's consider a point on the line AB with coordinates (0,y). We need to find the equation of the locus such that the distances from point P(3,0) and the line AB are equal.

Step 4: Use the distance formula:
The distance between two points (x1,y1) and (x2,y2) can be found using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). Using this formula, we can find the distance between point P(3,0) and the point (0,y). The distance between point P and the line AB is given by the perpendicular distance from point P to the line AB, which is the absolute difference between the x-coordinate of point P and the x-coordinate of the line AB.

Step 5: Set the distances equal to each other:
Since we want to find the equation of the locus where the distances from point P and the line AB are equal, we can set these distances equal to each other and solve for y.

Step 6: Simplify the equation:
By squaring both sides of the equation and simplifying, we can find the equation of the locus in terms of y.

Conclusion:
In conclusion, the equation of the locus of the point whose distances from point P(3,0) and the line AB are equal can be found by setting the distances equal to each other, solving for y, and simplifying the equation. The resulting equation