Q3 / 40 · NTA Abhyas (JEE Main) 2020 · Mathematics · Hyperbola

The locus of the mid-points of the chords of the hyperbola $3 x^{2} - 2 y^{2} + 4 x - 6 y = 0$ which are parallel to the line $y = 2 x + 4$ is

Correct answer
A
$3 x - 4 y = 4$
B
$4 x - 4 y = 3$
C
$3 y - 4 x + 4 = 0$
D
$3 x - 4 y = 2$

Solution

Let mid-point of the chord is

(h, k)

\Rightarrow

equation of the chord is

3 h x - 2 k y + 4 (\frac{x + h}{2}) - 6 (\frac{y + k}{2}) = 3 h^{2} - 2 k^{2} + 4 h - 6 k

\Rightarrow (3 h + 2) x + (- 2 k - 3) y = 3 h^{2} - 2 k^{2} + 2 h - 3 k

\Rightarrow

Slope of the chord is equal to

\frac{3 h + 2}{2 k + 3} = 2

\Rightarrow

locus of the mid-points is

3 x + 2 = 4 y + 6

\Rightarrow 3 x - 4 y - 4 = 0

Hence, the correct answer is option: 'A'.

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