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Question

The number of possible straight lines, passing through $$(2, 3)$$ and forming a triangle with coordinate axes, whose area is $$12 sq. units$$, is


A
One
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B
Two
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C
Three
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D
Four
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Solution

The correct option is C $$Three$$
Equation of any line through $$(2, 3)$$ is
$$y - 3 = m (x-2)$$
$$\Rightarrow y = mx - 2m +3$$
Given, area of $$\Delta$$OAB is $$\pm 12.$$ That is,
$$\displaystyle \frac{1}{2} \left ( \frac{2m-3}{m} \right ) (3- 2m) = \pm 12$$
Taking positive sign, we get $$(2m + 3)^2 = 0$$. This gives one value of m as 
$$\dfrac{-3}{2}$$. 

Taking negative sign, we get 
$$4m^2 - 36 m + 9 = 0 (D > 0)$$
This is a quadratic in $$m$$ which gives two real values of $$m.$$
Hence, three straight lines are possible.

Mathematics

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