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
B
Two
C
Three
D
Four

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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