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  • An ellipse slides between two straight lines at right angles to each other. The locus of its center is a :   Optio

An ellipse slides between two straight lines at right angles to each other. The locus of its center is a : 

 

Option: 1

Straight line
 


Option: 2

Ellipse 


Option: 3

Parabola
 


Option: 4

 Circle


Answers (1)

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Let the length of major and minor axes of an ellipse are 2 \mathrm{a} and \mathrm{2 \mathrm{~b}}  and if the center of the ellipse be \mathrm{\mathrm{C}(\mathrm{h}, \mathrm{k})}

If \mathrm{S\left(x_1, y_1\right) \, \, and\, \, S^{\prime}\left(x_2, y_2\right)}be two foci of the ellipse, then
\mathrm{\begin{array}{ll} & \mathrm{SS}^{\prime}=2 \mathrm{ae} \\ \Rightarrow \quad & \left(\mathrm{x}_1-\mathrm{x}_2\right)^2+\left(\mathrm{y}_1-\mathrm{y}_2\right)^2=4 \mathrm{a}^2 \mathrm{e}^2 \\ \Rightarrow & \\ & \\ & \left(\mathrm{x}_1+\mathrm{x}_2\right)^2-4 \mathrm{x}_1 \mathrm{x}_2+\left(\mathrm{y}_1+\mathrm{y}_2\right)^2-4 \mathrm{y}_1 \mathrm{y}_2=4\left(\mathrm{a}^2-\mathrm{b}^2\right) \\ \Rightarrow \quad & (2 \mathrm{~h})^2+(2 \mathrm{k})^2-4\left(\mathrm{x}_1 \mathrm{x}_2+\mathrm{y}_1 \mathrm{y}_2\right)=4\left(\mathrm{a}^2-\mathrm{b}^2\right) \ldots(1) \end{array}}
Since the ellipse always slides between the two fixed lines O X and O Y, they are always tangents to it. Therefore \mathrm{\mathrm{y}_1, \mathrm{y}_2 \, \, and\, \, \mathrm{x}_1, \mathrm{x}_2 }are perpendicular distances of the foci from these tangents, whose product are always b^2.
Hence \mathrm{x_1 x_2=y_1 y_2=b^2 }
\mathrm{\begin{aligned} & \Rightarrow \quad \text { Equation (1) becomes } 4 \mathrm{~h}^2+4 \mathrm{k}^2-8 \mathrm{~b}^2=4\left(\mathrm{a}^2-\mathrm{b}^2\right) \\ & \Rightarrow \quad \mathrm{h}^2+\mathrm{k}^2=\mathrm{a}^2+\mathrm{b}^2 \end{aligned} }
\mathrm{\therefore \quad Locus\, \, of\, \, the \, \, center (h, k) is\, \, the \, \, circle\, \, x^2+y^2=a^2+b^2. }

Posted by

Ritika Kankaria

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