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Question
Find the area of the triangle formed by the lines y – x = 0, x + y = 0 and x – k = 0.
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Solution
y – x = 0 and y + x = 0 meet at the point (0, 0).
By substituting x = k into y – x = 0, we get y – k = 0 or y = k
x – k = 0 and y – x = 0 meet at the point (k, k).
By substituting x = k into y + x = 0,
y + k = 0 or y = –k
x = k and y + x = 0 meet at the point (k, –k).
Now the area of the triangle formed by the points (0, 0), (k, k) and (k, –k) is
= `|1/2[0 xx (-2"k") + "k"(-"k") + "k" (-"k")]|`
= `|1/2 (-"k"^2 - "k"^2)|`
= k2 square units.
Second method: Area of triangle OPQ
= 2 × area ∆OAP
= `2 xx [1/2 xx "k" xx "k"]`
= k2 square units.
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