# The locus of the point of intersection of the straight lines,tx−2y−3t=0x−2ty+3=0  (t R), is : Option 1)  an ellipse with eccentricity Option 2)  an ellipse with the length of major axis 6   Option 3) a hyperbola with eccentricity Option 4)  a hyperbola with the length of conjugate axis 3

As we  learnt in

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.

Conjugate axis -

The line through the centre and perpendicular to transverse axis.

- wherein

$tx-2y-3t=0$

$x-2ty+3=0 (t\epsilon R)$

On solving, $x=\frac{-3(t^{2}+1)}{t^{2}-1}$

$y=\frac{3t}{(t^{2}-1)}$

$x^{2}-4y^{2}=\frac{9}{(t^{2}-1^{2})}\left [ (t^{2}+1)^{2} -t^{2}\right ]$

$x^{2}-4y^{2}=9$

$\frac{x^{2}}{9}-\frac{y^{2}}{(9/4)}=1$

Here a=3, b=$\frac{3}{2}$

Length of conjugrate axis

$=2b=3$

Option 1)

an ellipse with eccentricity

This option is incorrect

Option 2)

an ellipse with the length of major axis 6

This option is incorrect

Option 3)

a hyperbola with eccentricity

This option is incorrect

Option 4)

a hyperbola with the length of conjugate axis 3

This option is correct

N

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