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I am given a point $P(2,3)$ thru which passing line forms triangle with axes of area $12$ , so how many lines will pass thru $P$ making same area with axes?

Writing intercept form of line

$$\frac{x}{a} + \frac{y}{b}=1$$ and then satisfying point in this equation I get $$2b+3a=ab$$ , now since area is $12$ hence $$1/2ab=12$$ we get $ab=24$ now substituting $ab$ and $b$ i get $$a^2 -8a +14=0$$ . Now we have 2 values for $a & b$ hence 2 lines should pass thru that points satisfying condition but correct answer is 3 , what I did wrong?

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Of the three solutions, one triangle is in quadrant 1, one is in quadrant 2, and one is in quadrant 4.

You focused exclusively on a quadrant 1 triangle (from the assumption that $ab=24$. But you get solutions in the other quadrants if you look for solutions where $ab=-24$).

By the way for the first quadrant: From your equations $ab=24$ and $\frac2a+\frac3b=1$, you get $b=\frac{24}{a}$, which leads to $a^2-8a+16=0$. This quadratic has only one (double) root of $a=4$.

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