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  • An ellipse slides between straight lines y = 2x and 2y = -x then the locus of its centre is Option: 1 Straight line <div clas

An ellipse slides between straight lines y = 2x and 2y = -x then the locus of its centre is 

Option: 1

Straight line 


Option: 2

Circle


Option: 3

Parabola


Option: 4

Hyperbola


Answers (1)

best_answer

 

 

Director Circle -

Director Circle 

\\ \text {The equation of the director circle of the ellipse } \frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{b}^{2}}=1 \text { is } \\\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}

The locus of the point through which perpendicular tangents are drawn to a given Ellipse S = 0 is a circle called the director circle of the ellipse S = 0.

\\ {\text {Equation of tangent of the ellipse } \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text { in slope form is } y=m x+\sqrt{a^{2} m^{2}+b^{2}}} \\ {\text { it passes through the point }(h, k)} \\ {k=m h+\sqrt{a^{2} m^{2}+b^{2}}} \\ {(k-m h)^{2}=a^{2} m^{2}+b^{2}} \\ {k^{2}+m^{2} h^{2}-2 m h k=a^{2} m^{2}+b^{2}} \\ {\left(h^{2}-a^{2}\right) m^{2}-2 h k m+k^{2}-b^{2}=0} \\\\ {\text {This is quadratic equation in } \mathrm{m}, \text { slope of two tangents are } \mathrm{m}_{1} \text { and } \mathrm{m}_{2}} \\ {\mathrm{m}_{1} \mathrm{m}_{2}=\frac{\mathrm{k}^{2}-\mathrm{b}^{2}}{\mathrm{h}^{2}-\mathrm{a}^{2}}} \\ {-1=\frac{\mathrm{k}^{2}-\mathrm{b}^{2}}{\mathrm{h}^{2}-\mathrm{a}^{2}} \quad[\text { tangents are prpendicular }] } \\ {-\mathrm{h}^{2}+\mathrm{a}^{2}=\mathrm{k}^{2}-\mathrm{b}^{2}} \\ {\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2}+\mathrm{b}^{2}}

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given equation of two straight lines is y = 2x and 2y = -x, these lines intersects at 90o    

As ellipse is sliding between these two lines so locus of the center is the circle

Posted by

Kshitij

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