Show that the lines joining the origin to the other two points of intersection of the curves ax 2 +2hxy+by 2 +2gx = 0 and a ′ x 2 + 2h ′ xy + b ′ y 2 + 2g ′ x = 0 will be at right angles to one another if g(a ′ + b ′ ) = g ′ (a + b)
Text Solution
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ax 2 + 2hxy + by 2 + 2gx = 0 ....(1)
a ′ x 2 + 2h ′ xy + b ′ y 2 + 2g ′ x = 0 ....(2)
multiplying (1) by g ′ and (2) by g and subtracting we get
(ag ′ – a ′ g)x 2 + 2 (hg ′ – h ′ g)xy + (bg ′ – b ′ g)y 2 = 0 ⇒ for lines to be
(ag ′ – a ′ g) + (bg ′ – b ′ g) = 0 ⇒ g(a ′ + b ′ ) = g ′ (a + b)
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