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![[Post New]](/templates/default/images/icon_minipost_new.gif) 22 Apr 2008 22:05:33 IST
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ok..here is one more ppl... In  ABC the perpendicular bisectors of AB and AC are 3x + 2y - 5 = 0 and 2x - 3y + 1 = 0. If vertex A is ( 4,5 ) then distance between the orthocenter and circumcentre is ....
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22 Apr 2008 22:33:29 IST
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Its a BBBBBiggg problem if u solve it my way....
1. Solve the two equations given to get the circumcentre which = M(1,1)...find distance AM....5 units...
2.distance of BM = AM and distance of A from the bisector of AB is equal to dsitance of B from it.....using above two conditions solve for the point B....same way get pt C.
3.now we find equations of altitudes thru B and C....as v already know the slopes of these from the given equations of the bisectors...its only the constant term dat varies.....which is got by substituting pts B n C in the respective eqns.....
4. get the orthocentre solving the eqns of the altitudes.....now dat u kno the pts ....M (the circumcentre) n the ortho centre pt say N....find MN using distance formula.....
n eureka....soln got !!!
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22 Apr 2008 22:54:08 IST
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a little shorter method...
1.solve the given equations and find orthocentre that comes out to be 1,1.
2.thn assume the mid point of AB to be h,k..this will satisfy the given bisector eqn...u can also find the the coordinates of B in terms of h,k using midpoint formula...and the product of slope of orthocentre and mid point and AB =-1..
can u find B from this??
3. similarly finf C
4. u know A,B,C find centroid...thn G divides circumcentre and orthocentre in the ratio 1:2.. u can find circumcentre...
thn using distance formula u can find distance b/w them...
i hope u got it
its a long question..but by using the ratio u wud be saved from finding the equations of altitudes as suggested by greatnizzy
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22 Apr 2008 23:02:50 IST
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The Shortest method..... 1) since the perpendicular bisectors intersect at rt. angles, we can conclude angle a is 90 2) hence it is orthocentre 3)Find intersection of perpendicular bisectors and their distance with a Answer is 5
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22 Apr 2008 23:12:01 IST
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hey gr8 work harsh keep it up!!!!!!!!
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22 Apr 2008 23:19:17 IST
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thanx Aditi
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24 Apr 2008 10:28:31 IST
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Cool 1 dude !!!
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