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

Answer 1

The correct option is D four

Draw a line passing through $P(2,3)$

Let it cut at coordinate axis $A(x^{\prime },0)$ and $B(0,y^{\prime })$

The area of triangle $=12$

$\frac{\left|x^{\prime }{y}^{\prime }\right|}{2}=12$

So x' has two value

$x^{\prime }=\frac{24}{y^{\prime }}$ and $x^{\prime }=-\frac{24}{y^{\prime }}$

Now slope of $AP$ and $AB$ are equal

$\frac{y^{\prime }-3}{0-2}=\frac{y^{\prime }-0}{0-x^{\prime }}$

$2y^{\prime }+3x^{\prime }=x^{\prime }{y}^{\prime }$

putting both value of $x^{\prime }{y}^{\prime }$

$2y^{\prime }+\frac{72}{y^{\prime }}=24$

and

$2y^{\prime }-\frac{72}{y^{\prime }}=-24$

Hence both are quardratic eq SO possible solution is $4$