Question

Find the values of α so that the point P (α², α) lies inside or on the triangle formed by the lines x − 5y + 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0.

Answer 1

Let ABC be the triangle of sides AB, BC and CA whose equations are x − 5y + 6 = 0, x − 3y + 2 = 0 and x − 2y − 3 = 0, respectively.
On solving the equations, we get A (9,3), B (4, 2) and C (13, 5) as the coordinates of the vertices.


It is given that point P (α², α) lies either inside or on the triangle. The three conditions are given below.

(i) A and P must lie on the same side of BC.

(ii) B and P must lie on the same side of AC.

(iii) C and P must lie on the same side of AB.

If A and P lie on the same side of BC, then

$$\begin{gathered} \left(9-9+2\right)\left({\mathrm{\alpha }}^2-3\mathrm{\alpha }+2\right)\ge 0 \\ \Rightarrow \left(\mathrm{\alpha }-2\right)\left(\mathrm{\alpha }-1\right)\ge 0 \end{gathered}$$

$\Rightarrow \mathrm{\alpha }\in (-\infty ,1]\cup [2,\infty )$ ... (1)

If B and P lie on the same side of AC, then

$$\begin{gathered} \left(4-4-3\right)\left({\mathrm{\alpha }}^2-2\mathrm{\alpha }-3\right)\ge 0 \\ \Rightarrow \left(\mathrm{\alpha }-3\right)\left(\mathrm{\alpha }+1\right)\le 0 \end{gathered}$$

$\Rightarrow \mathrm{\alpha }\in \left[-1,3\right]$ ... (2)

If C and P lie on the same side of AB, then

$$\begin{gathered} \left(13-25+6\right)\left({\mathrm{\alpha }}^2-5\mathrm{\alpha }+6\right)\ge 0 \\ \Rightarrow \left(\mathrm{\alpha }-3\right)\left(\mathrm{\alpha }-2\right)\le 0 \end{gathered}$$

$\Rightarrow \mathrm{\alpha }\in \left[2,3\right]$ ... (3)

From (1), (2) and (3), we get:

$\mathrm{\alpha }\in \left[2,3\right]$