How many straight lines passing through (2,3) form a triangle with coordinate axes, whose area is 12 sq. units?

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Answer 1 · 2017-08-31T00:48:20

One

Let the equation of the straight line passing through $(2, 3)$ $(2,3)$ be $y - 3 = m (x - 3)$ $y-3=m(x-3)$ ,where $m$ $m$ represents the slope of the straight line.
Transforming the above equation in intercept form we have $\frac{x}{\frac{2 m - 3}{m}} + \frac{y}{3 - 2 m} = 1$ $x/((2m-3)/m)+y/(3-2m)=1$
So x-intercept $= \frac{2 m - 3}{m}$ $=(2m-3)/m$ and y-intercept $= (3 - 2 m)$ $=(3-2m)$
Area of the triangle formed by the straight line and intercepts from axes is given by
$\frac{1}{2} \times \frac{2 m - 3}{m} \times (3 - 2 m) = 12$ $1/2xx(2m-3)/mxx(3-2m)=12$

$\Rightarrow 4 m^{2} + 12 m + 9 = 0$ $=>4m^2+12m+9=0$

$\Rightarrow {(2 m + 3)}^{2} = 0$ $=>(2m+3)^2=0$

$\Rightarrow m = - \frac{3}{2}$ $=>m=-3/2$

As m has got only one real value, we can say that only one straight line passing through $(2, 3)$ $(2,3)$ is possible under the given condition.