Question

The number of possible straight lines, passing through \[(2,3)\] and forming a triangle with coordinates axes, whose area is \[12\;{\text{sq}}{\text{.units}}\] is-
A.One
B.Two
C.Three
D.Four

Answer 1

Hint: We are asked how many lines passing through the given point forms a triangle with coordinate axes such that the area of the triangle is \[12{\text{sq}}{\text{.units}}\] . For this such plot the point and line passing through the point. Recall the formula for the area of a triangle, use that and check how many possible lines can be formed with the given condition.

Complete step-by-step answer:
Given a point \[P(2,3)\]
Area of the triangle formed must be \[{\text{Area}} = 12\;{\text{sq}}{\text{.units}}\]

Let us plot the point and draw a line to understand properly,

Let AB be the line passing through the point \[P(2,3)\] . \[A(a,0)\] and \[B{\text{(}}0,b)\] are two points intersecting the x-axis and y-axis respectively.
We observe that for a triangle to be formed with the coordinate axes, the triangle must be a right angled triangle.
We know area of a right angled triangle can be written as,
 \[{\text{Area}} = \dfrac{1}{2} \times {\text{Base}} \times {\text{Height}}\] (i)
Here, we observe for triangle AOB, the base is \[OA = a\] and height is \[OB = b\] . Substituting these values in equation (i) we get,
 \[{\text{Area}} = \left| {\dfrac{1}{2} \times OA \times OB} \right|\]
 \[ \Rightarrow {\text{Area}} = \left| {\dfrac{1}{2} \times a \times b} \right|\] (ii)
We have an area to be equal to \[12\;{\text{sq}}{\text{.units}}\] . Substituting this value in equation (ii) we get,
 \[\left| {\dfrac{1}{2} \times a \times b} \right| = 12\;{\text{sq}}{\text{.units}}\]
 \[ \Rightarrow \left| {ab} \right| = 24\;{\text{sq}}{\text{.units}}\]
Values of \[a\] and \[b\] can be positive or negative. We now check how many such coordinates can be formed with both positive and negative values of \[a\] and \[b\] .
The possible values can be,
 \[(a,0)\] and \[(0,b)\]
 \[( - a,0)\] and \[(0,b)\]
 \[(a,0)\] and \[(0, - b)\]
 \[( - a,0)\] and \[(0, - b)\]
Therefore, four such sets can be formed which means four lines passing through the point \[(2,3)\] forms a triangle with the coordinate axes whose area is \[12\;{\text{sq}}{\text{.units}}\] .
So, the correct answer is “Option D”.

Note: The formulas for area of triangle, square, rectangle, cone, cylinder are very important. Most of the time students get confused with those formulas so remember properly the formulas for each shape. Whenever such questions are given where we need to find a set of possible values, first try to form a general equation and check for the possible values which will satisfy the general equation.