The orthocenter of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and 4x - y + 4 = 0 lies in
A
I quadrant
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B
II quadrant
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C
III quadrant
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D
IV quadrant
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Solution
The correct option is A I quadrant Coordinates of A and B are (-3, 4) and (−35,85) if orthocenter P(h, k) Then, (slope of PA)× (slope of BC) = - 1 k−4h+3×4=−1
⇒ 4k - 16 = -h - 3 ⇒ h + 4k = 13....(i)
and slope of PB× slope of AC = - 1 ⇒k−85h+35×−23=−1 ⇒5k−85h+3×23=1 ⇒ 10k - 16 = 15th + 9 15th - 10k + 25 = 10 3h - 2k + 5 = 0 ...(ii) Solving Eqs. (i) and (ii), we get h=37,k=227 Hence, orthocentre lies in I quadrant.