Question

If A = 7x + 5xy - 9y2 , B = -5x2 + 2xy + y2,C = 4x2 + 3y2 - 4xy, then find the value of A + B + C.​

Answer 1

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MATHS

Find the equation to the common conjugate diameters of the conics (1) x

2

+4xy+6y

2

=1 and 2x

2

+6xy+9y

2

=1, and (2) 2x

2

−5xy+3y

2

=1 and 2x

2

+3xy−9y

2

=1.

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ANSWER

To find the equation to the common conjugate diameters of the conics

1) The equations of the conic are

x

2

+4xy+6y

2

=1 .... (1) and 2x

2

+6xy+9y

2

=1 ... (2)

We know that, if the straight lines Ax

2

+2Hxy+By

2

are the conjugate diameters of ax

2

+2hxy+by

2

then, aB−2hH+bA=0

Now, let the straight lines Ax

2

+2Hxy+By

2

=0 ..... (3) be the conjugate diameters of both (1) and (2)

∴6A−4H+B=0 ........ (4)

and 9A−6H+2B=0 ...... (5)

Solving (4) and (5), we get

−8+6

A

=

9−12

H

=

−36+36

B

−2

A

=

−3

H

=

0

B

=k ( Say )

⇒A=−2k,H=−3k,B=0

Putting these values in (3), we have

−2kx

2

−6kxy=0 or k(x+3y)=0

Hence, (x+3y)=0