Question

The mid-point of the chord 4x - 3y = 5 of the hyperbola $2x^2-3y^2=12$ is

• A. \)
• B. \)
• C. \)
• D. \)

Answer 1

The correct option is B

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Let the mid-point of the chord is $(x_1,y_1)$

Now the equation of chord with given mid-point is

$T=S_1$

Here $S\equiv 2x^2-3y^2-12=0$

$T\equiv 2x{x}_1-3y{y}_1-12=0$

and $S_1\equiv 2\left(x_1^2\right)-3\left(y_1^2\right)-12=0$

$\Rightarrow$ $2x{x}_1-3y{y}_1-12=2x_1^2-3y_1^2-12$

$\Rightarrow$ $2x_{1}x-3y_{1}y=2x_1^2-3y_1^2$ ..............(1)

But the given equation of the chord is 4x-3y=5 ...........(2)

Both equation (1) $\&$ (2) represents same line ratio of their coefficients must be same

$\therefore \frac{2x_1}{4}=\frac{3y_1}{3}=\underset{}{\frac{2x_1^2-3y_1^2}{5}=k}$ (let's say)

$\Rightarrow$ $x_1=\frac{4k}{2}=2k$

$y_1=k$

$\frac{2x_1^2-3y_1^2}{5}=k$

$\frac{2(2k{)}^2-3(k{)}^2}{5}$

$2\times 4k^2-3k^2=5k$

$8k-3k=5$

$5k=5$

$k=1$

Then $x_1=2\times 1=2$

and $y_1=1$

coordinates of the mid-point of the chord is (2,1)