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1 Find thevalues of m for which x^2+3xy+x+my-m has two linear factors in x and y with integer coefficients 2 The length of the sides of a triangle are integrals and it's area is also integer.one side is 21 and the perimeter is 48. Find the smallest side.

1 Find thevalues of m for which x^2+3xy+x+my-m has two linear factors in x and y with integer coefficients
 
2 The length of the sides of a triangle are integrals and it's area is also integer.one side is 21 and the perimeter is 48. Find the smallest side.
 

Grade:12

2 Answers

Vijay Mukati
askIITians Faculty 2590 Points
8 years ago
Dear Student,

For Q1.

From the given two degree equation, we can write that:
x2+ 3xy + x +my -m = (ax+by+c)(dx+e)
Now multiply the two factors on RHS and equate the coefficient on both the sides.

On compairing you will directly get the value of m.

Thanks.
Vijay Mukati
askIITians Faculty 2590 Points
8 years ago
For Q2:

If we consider x to be the shortest side, then other sides will be27-x, 21
Now,x+21 > 27-x or x > 3. (Sum of any two sides is greater then the third side)
And x < 27 - x or x < 14 (Since x is the samllest side)
Therefore, 3 <x<14.

Now apply Heron's Formula.

s = (a+b+c)/2 = 48/2 = 24

A = sqrt(24(24-x)(x-3)(3))
A = sqrt(72(x-3)(24-x))
A = 6sqrt(2(x-3)(24-x))

The possible values of x are from 4 to 13.
Only for x = 10 will give an integeral area.
Therefore the answer is 10.

Thanks

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