Find the equation of the circle passing through the origin which is cu

1.	Find the equation of the circle passing through the origin which is cutting the chord of equal length √2 on the lines y = x and y = −x.
Answer» Let S ≡ x² + y² + 2gx + 2fy + c = 0 be the required circle. Since it passes through the origin (0, 0), c = 0. Put y = x in S = 0. Then x² + (g + f )x = 0. Therefore, x = 0, x = −(g + f ) so that the points of intersection are A(0, 0) and B[−(f + g), −(f + g)]. Now AB = √2 f + g = ±1 Similarly g - f = ± Therefore, the centres of circles are given by (1, 0), (−1, 0), (0, 1) and (0, −1) and the equations of the circle are given by + x² + y² ± 2x = 0 and x² + y² ± 2y = 0.

Find the equation of the circle passing through the origin which is cutting the chord of equal length √2 on the lines y = x and y = −x.

Answer»

Let S ≡ x² + y² + 2gx + 2fy + c = 0 be the required circle. Since it passes through the origin (0, 0), c = 0. Put y = x in S = 0. Then x² + (g + f )x = 0. Therefore, x = 0, x = −(g + f ) so that the points of intersection are A(0, 0) and B[−(f + g), −(f + g)]. Now

AB = √2

f + g = ±1

Similarly

g - f = ±

Therefore, the centres of circles are given by (1, 0), (−1, 0), (0, 1) and (0, −1) and the equations of the circle are given by + x² + y² ± 2x = 0 and x² + y² ± 2y = 0.

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Find the equation of the circle passing through the origin which is cutting the chord of equal length √2 on the lines y = x and y = −x.

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