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Question

The angle between the lines represented by the equation ax2+xy+by2=0 will be 45°, if


A

a=1,b=6

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B

a=1,b=-6

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C

a=6,b=1

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D

None of these

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Solution

The correct option is B

a=1,b=-6


Explanation for correct option:

Finding the values of a,b:

Given the equation ax2+xy+by2=0

Now comparing the general form of an equation of a pair of line ax2+2hxy+by2=0

We get a=a,b=b,2h=1h=12

And the given angle is 45°

Now the angle between the lines is given by

tanθ=2h2-aba+b

Substituting h=12 and θ=45°in the above equation we get,

tan45°=2122-aba+b1=214-aba+b(i)tan45°=1

Let us check with the options

Option (A): a=1,b=6

Substitute a=1,b=6 in equation (i)

L.H.S=1R.H.S=214-aba+b=214-1×61+6=21-2447=27×-232=-237L.H.SR.H.S

Therefore, option(A) is not correct

Option (B): a=1,b=-6

Substitute a=1,b=6 in equation (i)

L.H.S=1R.H.S=214-aba+b=214-1×-61-6=21+244-5=2-5×252=5-5=-1=1L.H.S=R.H.S

Therefore, option(B) is not correct

Option (C): a=6,b=1

Substitute a=6,b=1 in equation (i)

L.H.S=1R.H.S=214-aba+b=214-1×61-6=21-244-5=2-5×-232=-23-5L.H.SR.H.S

Therefore, option(C) is not correct

Option (D): None of these

This is not correct because option (B) satisfies the condition.

Therefore, only option (B) satisfies the given condition.

Hence, option (B) is the correct answer.


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