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  • The locus of the middle points of chords of hyperbola <img alt="\mathrm{3 x^2-2 y^2+4 x-6 y=0}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B3%20x%5E2

The locus of the middle points of chords of hyperbola \mathrm{3 x^2-2 y^2+4 x-6 y=0} parallel to y = 2x is

 

Option: 1

3x − 4y = 4


Option: 2

3y − 4x + 4 = 0


Option: 3

4x − 4y = 3


Option: 4

3x − 4y = 2

 


Answers (1)

best_answer

Let (h, k) be the mid-point of a chord of the hyperbola \mathrm{3 x^2-2 y^2+4 x-6 y=0 .}

Then the equation of the chord is

\mathrm{\begin{array}{lll} & \left.3 h k-2 k y+2(x+h)-3(y+k)=3 h^2-2 k^2+4 h-6 k \quad \text { [using } T=S^{\prime}\right] \\ \Rightarrow \quad & (3 h+2) x-(2 k+3) y-3 h^2+3 h^2+2 k^2-2 h+3 k=0 . & \end{array}}

This is parallel to y = 2x, therefore \mathrm{\frac{-(3 h+2)}{-(2 k+3)}=2}

⇒ 3h + 2 = 4k + 6   ⇒  3h − 4k = 4

Hence, locus of (h, k) is 3x − 4y = 4

 

Posted by

Ajit Kumar Dubey

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