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प्रश्न
The locus of the mid-point of the portion intercepted between the axes of the variable line x cos α + y sin α = p, where p is a constant, is
पर्याय
x2 + y2 = 4p2
`1/x^2 + 1/y^2 = 4/"p"^2`
`x^2 + y^2 = 4/"p"^2`
`1/x^2 + 1/y^2 = 2/"p"^2`
उत्तर
`1/x^2 + 1/y^2 = 4/"p"^2`
Explanation:
The straight line x cos α + y sin α = p meets the X-axis at the point A `("p"/(cos alpha), 0)` and the Y-axis at the point B`(0, "p"/(sin alpha))`.
Let (h, k) be the co-ordinates of the middle point of the line segment AB.
Then, h = `"p"/(2 cos alpha)` and k = `"p"/(2 sin alpha)`
⇒ cos α = `"p"/"2h" and sin alpha = "p"/"2k"`
cos2 α + sin2 α = `"p"^2/"4h"^2 + "p"^2/"4k"^2`
`=> 1 = "p"^2/4 (1/"h"^2 + 1/"k"^2)`
`=> 1/"h"^2 + 1/"k"^2 = 4/"p"^2`
Hence locus of the point (h, k) is
`1/x^2 + 1/y^2 = 4/"p"^2`
