Question

For the curve 7x²- 2y²+12xy - 2x + 14y -22 =0 which of the following is true ?
A) It is an hyperbola with eccentricity √3
B) It is an hyprbola with directrix 2x+y-1=0
C) It is an hyperbola with focus (1,2)
D) All​

Answer 1

hey frnd here is your answer

for the curve 7x*2 -2y*2+12xy-2x+14y-22=0

option C is the answer ..

hope helps you ...

Answer 2

Step-by-step explanation:

So the equation is

7x^2 – 2y^2 + 12xy – 2x + 14y – 22 = 0

Now we need to complete the square.

So (x + y + a)^2 = x^2 + y^2 + a^2 + 2xy +2ax + 2ay

By trial and error method

[ - 3(2x + y – 1)^2 = - 3(4x^2 + y^2 + 1 + 4xy – 4x - 2y)

                         = - 12x^2 – 3y^2 – 3 – 12 xy + 12x + 6y

-3(2x + y – 1)^2 + 12x^2 + 3y^2 + 3 – 12x – 6y = - 12xy]

So we get

  = - 3(2x + y – 1)^2 + 12x^2 + 3y^2 + 3 – 12x – 6y + 2x – 14y + 22

So – 5x^2 – 5y^2 + 10x + 20y – 25 = - 3(2x + y – 1)^2

       5x^2 + 5y^2 – 10x + 25 = 3(2x + y – 1)^2

Now to complete the square we get

    5(x^2 + y^2 – 2x – 4y + 5) = 3(2x + y – 1)^2

  So x^2 – 2.1.x + 1^2  + y^2 – 2.2y + 2^2 – 2^2 + 5 = 3/5 (2x + y – 1)^2

        So we get

              (x – 1)^2 + (y – 2)^2 = 3/5 (2x + y – 1)^2

    Taking square root we get

 Sq root (x – 1)^2 + (y – 2)^2 = sq root 3/5 (2x + y – 1)

We have the distance formula sq root (x1 – x2)^2 + (y1 – y2)^2

If a line be ax + by + c = 0 and point (x1,y1) then distance will be ax1 + by1 + c / root a^2 + b^2

Now there is a line 2x + y – 1 = 0

Now its distance will be m(x,y) and the point from this will be p(1,2) and let the fixed point be o.

So the ratio of mp and mo will be the eccentricity

So eccentricity = root 3 > 1

The directrix will be 2x + y – 1 = 0

Also the point p(1,2) is the focus.

Therefore option D All holds good.

Reference link will be

https://brainly.in/question/13660098