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The locus of the point of intersection of the straight lines tx-2y-3t=0 x-2ty+3=0 t€R. Is (1) an ellipse with eccentricity 2/√5 (2) an ellipse with the length of major axis 6 (3) a hyperbola with eccentricity √5 (4) a hyperbola with the length of conjugate axis 3 Please help

 
The locus of the point of intersection of the straight lines
tx-2y-3t=0
x-2ty+3=0
t€R.      Is
 
(1) an ellipse with eccentricity 2/√5
(2) an ellipse with the length of major axis 6
(3) a hyperbola with eccentricity √5
(4) a hyperbola with the length of conjugate axis 3
 
Please help

Grade:12

1 Answers

Arun
25750 Points
5 years ago
Dear student
 
From equation 1
 
t = 2y/(x- 3)
 
put this in equation 2
 
x – 4 y2 / (x – 3)  + 3 = 0
 
x^2 – 3x – 4y^2 + 3x – 9 = 0
 
x^2 – 4y^2 = 9
 
now you can check this as it is a hyperbola
 
a = 3, b = 3/2 
 
hence conjugate axis = 2b = 3
e = sqrt (1 + 1/4)
 
e = sqrt (5) /2
 

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