The orthocenter of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and

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The orthocenter of the triangle formed by the lines x + y = 1, 2x + 3y = 6 and 4x - y + 4 = 0 lies in

Options:

A . I quadrant

B . II quadrant

C . III quadrant

D . IV quadrant

Answer: Option A
:
A
Coordinates of A and B are (-3, 4) and

(- \frac{3}{5}, \frac{8}{5})

if orthocenter P(h, k)

The orthocenter Of The Triangle Formed By The Lines X + Y =...

Then, (slope of PA)

\times

(slope of BC) = - 1

\frac{k - 4}{h + 3} \times 4 = - 1

\Rightarrow

4k - 16 = -h - 3

\Rightarrow

h + 4k = 13....(i)
and slope of PB

\times

slope of AC = - 1

\Rightarrow \frac{k - \frac{8}{5}}{h + \frac{3}{5}} \times - \frac{2}{3} = - 1

\Rightarrow \frac{5 k - 8}{5 h + 3} \times \frac{2}{3} = 1

\Rightarrow

10k - 16 = 15th + 9
15th - 10k + 25 = 10
3h - 2k + 5 = 0 ...(ii)
Solving Eqs. (i) and (ii), we get

h = \frac{3}{7}, k = \frac{22}{7}

Hence, orthocentre lies in I quadrant.

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