A variable line L passing through the point B(2,5) intersects the lines 2x^2 - 5xy + 2y^2=0 at P and Q. Find the locus of the point R on L such thatthe distances BP BR and BQ are in harmonic progression

To find the locus of the point R on the line L such that the distances BP, BR, and BQ are in harmonic progression, we first need to understand the geometric implications of harmonic progression and how it relates to the distances involved.

Understanding Harmonic Progression

Three distances are said to be in harmonic progression if the reciprocals of these distances are in arithmetic progression. In our case, we have distances BP, BR, and BQ. Therefore, we can express this condition mathematically as:

• Let BP = a
• Let BR = b
• Let BQ = c

The condition for these distances to be in harmonic progression can be written as:

2/BR = 1/BP + 1/BQ

Rearranging gives us:

BR^2 = BP * BQ

Setting Up the Problem

Given that the line L passes through the point B(2, 5), we can express the equation of line L in slope-intercept form as:

y - 5 = m(x - 2),

where m is the slope of the line. Rearranging this gives:

y = mx - 2m + 5.

Finding Points of Intersection

The equation of the curve given is:

2x^2 - 5xy + 2y^2 = 0.

This can be factored or solved using the quadratic formula to find the points P and Q where line L intersects the curve. To do this, we can substitute the expression for y from the line equation into the curve equation:

2x^2 - 5x(mx - 2m + 5) + 2(mx - 2m + 5)^2 = 0.

Solving this quadratic equation will yield the x-coordinates of points P and Q. Let's denote these x-coordinates as x1 and x2, corresponding to points P and Q respectively.

Calculating Distances

Next, we need to calculate the distances BP and BQ:

BP = √[(x1 - 2)² + (y1 - 5)²]

BQ = √[(x2 - 2)² + (y2 - 5)²]

Now, we can express BR in terms of the coordinates of R, which lies on line L:

Let R have coordinates (x, mx - 2m + 5). Then:

BR = √[(x - 2)² + (mx - 2m + 5 - 5)²] = √[(x - 2)² + (mx - 2m)²].

Establishing the Locus Condition

Substituting BP, BR, and BQ into the harmonic progression condition:

BR² = BP * BQ.

After substituting the expressions for BP, BR, and BQ, we will arrive at a relationship involving x and m. This relationship will define the locus of point R.

Final Steps

After simplifying the equation, we can express it in a standard form, which will typically be a conic section (like a parabola, ellipse, or hyperbola) depending on the nature of the original curve and the line L.

In conclusion, the locus of point R can be derived from the distances and their relationships, ultimately leading to a specific geometric shape that represents all possible positions of R under the given conditions. This process involves algebraic manipulation and understanding of geometric properties, which is crucial in solving such problems in coordinate geometry.