if the circle x^2 +y^2 −2y−8=0 and x^2 +y^2 −24x+hy=0 cut orthogonally, determine the value of h.


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Question Number 37591 by mondodotto@gmail.com last updated on 15/Jun/18

$if the circle x^{2} + y^{2} - 2 y - 8 = 0$ $and x^{2} + y^{2} - 24 x + h y = 0 cut orthogonally,$ $determine the value of h .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 15/Jun/18

	$x^{2} + y^{2} - 2 y - 8 = 0$ $c o m p a r i n g w i t h x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ $c e n t r e (- g, - f) r a d i u s = \sqrt{g^{2} + f^{2} - c}$ $o_{1} (0, 1) r_{1} = \sqrt{0^{2} + {(- 1)}^{2} - (- 8)} = \sqrt{9} = 3$ $o_{2} (12, - \frac{h}{2}) r_{2} = \sqrt{{(- 12)}^{2} + {(\frac{h}{2})}^{2}}$ $r_{2} = \sqrt{144 + \frac{h^{2}}{4}}$ $o_{1} o_{2} = \sqrt{{(12 - 0)}^{2} + {(- \frac{h}{2} - 1)}^{2}}$ $= \sqrt{144 + \frac{h^{2}}{4} + h + 1} = \sqrt{\frac{h^{2}}{4} + h + 145}$ $r_{1}^{2} + r_{2}^{2} = {(o_{1} o_{2})}^{2}$ $9 + 144 + \frac{h^{2}}{4} = \frac{h^{2}}{4} + h + 145$ $h = 8$ $o r m d t h o d$ $c l n d i t i o n f o r o r t h o g o n s l i t y$ $2 g_{1} g_{2} + 2 f_{1} f_{2} = c_{1} + c_{2}$ $2 \times 0 \times (- 12) + 2 \times (- 1) \times \frac{h}{2} = - 8 + 0$ $- h = - 8$ $h = 8$
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