Question -
Answer -
(i) x2┬а+ xy тАУ 3x тАУ y + 2 = 0
Firstly let us substitute the value of x by x + 1 and y by y + 1
Then,
(x + 1)2┬а+ (x +1) (y + 1) тАУ 3(x + 1) тАУ (y + 1) + 2 = 0
x2┬а+ 1 + 2x + xy+ x + y + 1 тАУ 3x тАУ 3 тАУ y тАУ 1 + 2 = 0
Upon simplification we get,
x2┬а+ xy = 0
тИ┤ Thetransformed equation is x2┬а+ xy= 0.
(ii) x2┬атАУy2┬атАУ 2x + 2y = 0
Let us substitute the value of x by x + 1 and y by y + 1
Then,
(x + 1)2┬атАУ (y + 1)2┬атАУ 2(x + 1) + 2(y + 1) = 0
x2┬а+ 1 + 2x тАУ y2┬атАУ 1 тАУ 2y тАУ 2x тАУ 2 + 2y + 2 = 0
Upon simplification we get,
x2┬атАУ y2┬а= 0
тИ┤ Thetransformed equation is x2┬атАУ y2┬а= 0.
(iii) xy тАУ x тАУ y + 1 = 0
Let us substitute the value of x by x + 1 and y by y + 1
Then,
(x + 1) (y + 1) тАУ (x + 1) тАУ (y + 1) + 1 = 0
xy + x + y + 1 тАУ x тАУ 1 тАУ y тАУ 1 + 1 = 0
Upon simplification we get,
xy = 0
тИ┤ Thetransformed equation is xy = 0.
(iv) xy тАУ y2┬атАУx + y = 0
Let us substitute the value of x by x + 1 and y by y + 1
Then,
(x + 1) (y + 1) тАУ (y + 1)2┬атАУ(x + 1) + (y + 1) = 0
xy + x + y + 1 тАУ y2┬атАУ1 тАУ 2y тАУ x тАУ 1 + y + 1 = 0
Upon simplification we get,
xy тАУ y2┬а= 0
тИ┤ Thetransformed equation is xy тАУ y2┬а=0.
4. At what point the origin beshifted so that the equation x2┬а+xy тАУ 3x + 2 = 0 does not contain any first-degree term and constant term?
Solution:
Given:
The equation x2┬а+xy тАУ 3x + 2 = 0
We know that the origin has been shifted from (0, 0) to (p, q)
So any arbitrary point (x, y) will also be converted as (x + p,y + q).
The new equation is:
(x + p)2┬а+ (x +p)(y + q) тАУ 3(x + p) + 2 = 0
Upon simplification,
x2┬а+ p2┬а+ 2px + xy + py + qx + pq тАУ 3x тАУ 3p+ 2 = 0
x2┬а+ xy + x(2p +q тАУ 3) + y(q тАУ 1) + p2┬а+ pq тАУ 3pтАУ q + 2 = 0
For no first degree term, we have 2p + q тАУ 3 = 0 and p тАУ 1 = 0,and
For no constant term we have p2┬а+pq тАУ 3p тАУ q + 2 = 0.
By solving these simultaneous equations we have p = 1 and q = 1from first equation.
The values p = 1 and q = 1 satisfies p2┬а+pq тАУ 3p тАУ q + 2 = 0.
Hence, the point to which origin must be shifted is (p, q) = (1,1).