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The locus of the point of intersection of the lines,\sqrt2x-y+4\sqrt2k = 0 and \sqrt2Kx+Ky-4\sqrt2k = 0    (k is any non-zero real parameter), is :

 

  • Option 1)

    an ellipse whose eccentricity is \frac{1}{\sqrt3}

  • Option 2)

    an ellipse with length of its major axis8\sqrt2

  • Option 3)

    a hyperbola whose eccentricity is \sqrt3 

  • Option 4)

    a hyperbola with length of its transverse axis8\sqrt2

 

Answers (1)

best_answer

As we have learned

Locus -

Path followed by a point p(x,y) under given condition (s).

- wherein

It is satisfied by all the points (x,y) on the locus.

 

 

Transverse axis -

The line through the foci of the hyperbola.

- wherein

 

 

Hyperbola -

Hyperbola is locus of all the points in a plane ,the difference of whose distance from two fixed point is constant.

- wherein

 

\sqrt2x-y= -4\sqrt2k

k(\sqrt2x+y)= 4\sqrt2

also (\sqrt2x+y)(\sqrt2x-y)= -16\cdot 2

2x^{2}-y^{2}=-32

\frac{y^{2}}{32}-\frac{x^{2}}{16}=1

this is hyperbola  a^{2}=32 \Rightarrow a=4\sqrt2

transverse  axis = 2a = 8\sqrt2

 

 

 

 


Option 1)

an ellipse whose eccentricity is \frac{1}{\sqrt3}

This is incorrect

Option 2)

an ellipse with length of its major axis8\sqrt2

This is incorrect

Option 3)

a hyperbola whose eccentricity is \sqrt3 

This is incorrect

Option 4)

a hyperbola with length of its transverse axis8\sqrt2

This is correct

Posted by

Himanshu

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