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Home » Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.
Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.
CBSE | Class 11 | Exercise 23.14 | Maths | RD Sharma | Straight Lines
Find the values of α so that the point P(α 2, α) lies inside or on the triangle formed by the lines x – 5y + 6 = 0, x – 3y + 2 = 0 and x – 2y – 3 = 0.

According to ques,:

    \[x\text{ }\text{ }5y\text{ }+\text{ }6\text{ }=\text{ }0,\]

    \[x\text{ }\text{ }3y\text{ }+\text{ }2\text{ }=\text{ }0\]

and

    \[x\text{ }\text{ }2y\text{ }\text{ }3\text{ }=\text{ }0\]

forming a triangle and point P(α2, α) lies inside or on the triangle

Let ABC be the triangle of sides AB, BC and CA whose equations are:

    \[~x~-~5y\text{ }+\text{ }6\text{ }=\text{ }0,\]

    \[x~-~3y\text{ }+\text{ }2\text{ }=\text{ }0\]

and

    \[x~-~2y~-~3\text{ }=\text{ }0,\]

respectively.

On solving the equations, we have A (9, 3), B (4, 2) and C (13, 5) as the coordinates of the vertices.

RD Sharma Solutions for Class 11 Maths Chapter 23 – The Straight Lines - image 70

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

    \[\left( 9\text{ }\text{ }9\text{ }+\text{ }2 \right)\left( {{\alpha }^{2}}~\text{ }3\alpha ~+\text{ }2 \right)~\ge 0\]

    \[\left( \alpha ~\text{ }2 \right)\left( \alpha ~\text{ }1 \right)~\ge ~0\]

    \[\alpha ~\in ~\left( -~\infty ,\text{ }1\text{ } \right]~\cup ~\left[ \text{ }2,~\infty \right)\text{ }\ldots \text{ }\left( 1 \right)\]

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

    \[\left( 4\text{ }\text{ }4\text{ }\text{ }3 \right)\text{ }\left( {{\alpha }^{2}}~\text{ }2\alpha ~\text{ }3 \right)~\ge ~0\]

    \[\left( \alpha ~\text{ }3 \right)\left( \alpha ~+\text{ }1 \right)~\le ~0\]

    \[\alpha ~\in ~\left[ -\text{ }1,\text{ }3 \right]\text{ }\ldots \text{ }\left( 2 \right)\]

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

    \[\left( 13\text{ }\text{ }25\text{ }+\text{ }6 \right)\left( {{\alpha }^{2}}~\text{ }5\alpha ~+\text{ }6 \right)~\ge 0\]

    \[\left( \alpha ~\text{ }3 \right)\left( \alpha ~\text{ }2 \right)~\le ~0\]

    \[\alpha ~\in ~\left[ \text{ }2,\text{ }3 \right]\text{ }\ldots \text{ }\left( 3 \right)~~~\]

From equations (1), (2) and (3), we have

    \[\alpha \in ~\left[ 2,\text{ }3 \right]\]

    \[\therefore \alpha \in ~\left[ 2,\text{ }3 \right]\]

i

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