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Geometry Level pending

Find a line which passes through the centres of all the circles passing through (3,2) and(4,7).

2 x + y − 23 = 0 2x +y -23 = 0 x + 2 y − 25 = 0 x + 2y -25=0 x + y − 16 = 0 x + y - 16 = 0 2 x + 10 y − 52 = 0 2x + 10y - 52 = 0

Arnab Das by Arnab Das , 24, India

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2 solutions

Arnab Das
Oct 1, 2014

Let us say the centre is (h,k) . The distance from the centres to (3,2) and(4,7) would be equal ( and equal to radius) so ( h − 3 ) 2 + ( k − 2 ) 2 = ( h − 4 ) 2 + ( k − 7 ) 2 (h-3)^{2} + (k-2)^{2} = (h-4)^{2} + (k-7)^{2}

On solving we get 2 h + 10 k − 52 = 0 2h + 10k -52 = 0

On replacing with variables x and y we get 2 x + 10 y − 52 = 0 2x +10y -52 = 0

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Michael Mendrin
Sep 29, 2014

The slope between (3,2) and (4,7) is 5, so the line in question has to have a slope of -(1/5), and only one of the choices fit.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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