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Question

Find the values of α so that the point P(α2,α) lies inside or on the triangle formed by the lines x5y+6=0, x3y+2=0 and x2y3=0

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Solution

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

x5y+6=0, x3y+2=0 and x2y3=0

On solving equations, we get

A(9, 3), B(4, 2) and C(13, 5)

If the point P(α,α2) lies n side the ΔABC, then

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

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

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

Now,

A and P are on the same side of BC if

{9(1)+3(3)+2}(α23α+2)>0

(99+2)(α23α+2)>0

α23α+2>0

(α1)(α2)>0

αϵ(,1)v(2,)(i)

B and P will lie on the same side of CA, if

{13(1)+5(5)+6}(α25α+6)>0

(6)(α25α+6)<0

α25α+6<0

(α2)(α3)<0

αϵ(2,3)(ii)

C and P will lie on the same side of AB, if

{4(1)+2(2)3}(α22α3)>0

(3)(α22α3)>0

α22α3<0

(α3)(α+1)<0

αϵ(1,3)(iii)

From (i), (ii), (iii)

αϵ[2,3]


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