The locus of the mid-point of the chord of th | vedclass

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Question

(A) Let $(h, k)$ be the mid-point of a chord of the hyperbola $S: 3x^2 - 2y^2 + 4x - 6y = 0$.The equation of the chord with mid-point $(h, k)$ is give

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$3x - 4y = 4$

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(A) Let $(h, k)$ be the mid-point of a chord of the hyperbola $S: 3x^2 - 2y^2 + 4x - 6y = 0$.The equation of the chord with mid-point $(h, k)$ is given by $T = S_1$,where $T$ is the tangent-like expression and $S_1$ is the value of the expression at $(h, k)$.$3hx - 2ky + 2(x + h) - 3(y + k) = 3h^2 - 2k^2 + 4h - 6k$Rearranging the terms to find the slope:$(3h + 2)x - (2k + 3)y = 3h^2 - 2k^2 + 2h - 3k$The slope of this chord is $m = \frac{3h + 2}{2k + 3}$.Since the chord is parallel to $y = 2x$,its slope must be $2$.$\frac{3h + 2}{2k + 3} = 2$$3h + 2 = 4k + 6$$3h - 4k = 4$Replacing $(h, k)$ with $(x, y)$,the locus is $3x - 4y = 4$.

The locus of the mid-point of the chord of the hyperbola $3x^2 - 2y^2 + 4x - 6y = 0$ which is parallel to $y = 2x$ is:

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