The locus of the middle points of the chords of hyperbola 3 x^2 -… - Vidyadip

MCQ

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

✓
$3x - 4y = 4$
B
$3y - 4x + 4 = 0$
C
$4x - 4y = 3$
D
$3x - 4y = 2$

✓

Answer

Correct option: A.

$3x - 4y = 4$

a
(a) Let $P({x_1},{y_1})$be the middle point of the chord of the

hyperbola $3{x^2} - 2{y^2} + 4x - 6y = 0$

$\therefore$ Equation of the chord is $T = {S_1}$

i.e. $3x{x_1} - 2y{y_1} + 2(x + {x_1}) - 3(y + {y_1}) = 0$

or $(3{x_1} + 2)x - (2{y_1} + 3)y + (2{x_1} - 3{y_1}) = 0$

If this chord is parallel to line $y = 2x,$ then

${m_1} = {m_2}$==> $ - \frac{{3{x_1} + 2}}{{ - (2{y_1} + 3)}} = 2$ ==>$3{x_1} - 4{y_1} = 4$

Hence the locus of the middle point $({x_1},{y_1})$ is $3x -4y=4$.

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