First slide

Cartesian plane

Question

The orthocentre of the triangle formed by the lines $y = 0, (1 + t) x - t y + t (1 + t) = 0$ and $(1 + u) x - u y$ $+ u (1 + u) = 0 (t \neq u)$ for all values of $t$ and $u$ lies on the line.

Moderate

A

$x - y = 0$
B

$x + y = 0$
C

$x - y + 1 = 0$
D

$x + y + 1 = 0$

Solution

The orthocentre $(x, y)$ lies on the lines.
$y = \frac{- u}{1 + u} (x + t)$
and $y = \frac{- t}{1 + t} (x + u)$
$\begin{array}{l} \Rightarrow [(1 + u) - (1 + t)] y + (u - t) x = 0 \\ \Rightarrow (u - t) (x + y) = 0 \\ \Rightarrow x + y = 0 as u \neq t \end{array}$

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