Question

image of the curve x^2+ y^2=1 in the line x + y=1​

Answer 1

we have to find the image of the curve x² + y² = 1 in the line x + y = 1.

here we know, x² + y² = 1 is equation of circle where centre (0,0) and radius 1 unit.

let centre of image of curve x² + y² = 1 is (h, k)

see figure,

slope of( x + y = 1 ) × slope of joining the points (h, k) and (0,0) = -1

⇒-1 × (k - 0)/(h - 0) = -1

⇒k = h ......(1)

as distance of (0,0) from line x + y = 1 is same as distance of (h, k) from the line x + y = 1.

so, (h/2, k/2) lies on x + y = 1

then, h/2 + k/2 = 1 ⇒h + k = 2

from equation (1) we get, (h, k) = (1, 1)

now equation of circle of centre (1,1) and radius 1 unit .

so, (x - 1)² + (y - 1)² = 1²

⇒x² + y² - 2x - 2y + 1 = 0

therefore x² + y² - 2x - 2y + 1 is image of given curve.