If x2 – y2 + 2hxy + 2gx + 2fy + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines x + 2y + 7 = 0 and 2x – y + 8 = 0, then the value of g + c + h

Engineering

Mathematics

Length of Perpendicular

Question

If x² – y² + 2hxy + 2gx + 2fy + c = 0 is the locus of a point, which moves such that it is always equidistant from the lines x + 2y + 7 = 0 and 2x – y + 8 = 0, then the value of g + c + h – f equals.

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$\frac{| 2 h - k + 8 |}{\sqrt{5}} = \frac{| h + 2 k + 7 |}{\sqrt{5}}$

4h² + k² + 64 – 4hk + 32h – 16k

= h² + 4k² + 49 + 4kh + 14h + 28k

3h² – 3k² – 8hk+ 18h – 44k + 15 = 0

$x^{2} - y^{2} - \frac{8}{3} xy + 6 x - \frac{44}{3} y + 5 = 0$

$g + c + h - f = 3 + 5 - \frac{4}{3} + \frac{22}{3}$

= 8 + 6 = 14