SATHEE: Straight Line and Pair of Straight Lines 1 Question 45

Straight Line and Pair of Straight Lines 1 Question 45

45. The orthocentre of the triangle formed by the lines $x + y = 1, 2 x + 3 y = 6$ and $4 x - y + 4 = 0$ lies in quadrant number… .

$(1985, 2 M)$

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Solution:

Let $H (h, k)$ be orthocentre.

$\Rightarrow ($ slope of $A H) \cdot ($ slope of $B C) = - 1$

$\Rightarrow \frac{k - \frac{16}{7}}{h + \frac{3}{7}} \cdot (- 1) = - 1$

$\begin{array}{lc} \Rightarrow & k - \frac{16}{7} = h + \frac{3}{7} \\ \Rightarrow & h - k = - \frac{19}{7} \end{array}$

Also, $($ slope of $C H) \cdot ($ slope of $A B) = - 1$

$\begin{aligned} \Rightarrow & \frac{k - 4}{h + 3} \cdot (4) & = - 1 \\ \Rightarrow & 4 k - 16 & = - h - 3 \\ \Rightarrow & h + 4 k & = 13 \end{aligned}$

On solving Eqs. (i) and (ii), we get $h = \frac{3}{7}, k = \frac{22}{7}$

$∴$ Orthocentre $\frac{3}{7}, \frac{22}{7}$

Hence, this coordinate lies in the first quadrant.

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Straight Line and Pair of Straight Lines 1 Question 45

45. The orthocentre of the triangle formed by the lines x+y=1,2x+3y=6 and 4x−y+4=0 lies in quadrant number… .

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45. The orthocentre of the triangle formed by the lines $x + y = 1, 2 x + 3 y = 6$ and $4 x - y + 4 = 0$ lies in quadrant number… .