A line of fixed length a + b moves so that its ends are always on two fixed perpendicular straight lines. Then the locus of a point which divides this line into portions of length a and b is

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an ellipse

a parabola

a straight line

a hyperbola

answer is A.

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Detailed Solution

$AB = a + b \Rightarrow {AB}^{2} = {(a + b)}^{2} α^{2} + β^{2} = {(a + b)}^{2} \to (1) P (x, y) = (\frac{bα}{a + b}, \frac{aβ}{a + b}) α = \frac{(a + b) x}{b}, β = \frac{(a + b) y}{a} F r o m (1) \frac{{(a + b)}^{2} x^{2}}{b^{2}} + \frac{{(a + b)}^{2} y^{2}}{a^{2}} = {(a + b)}^{2} \Rightarrow \frac{x^{2}}{b^{2}} + \frac{y^{2}}{a^{2}} = 1$

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