Two chords of the circle x^2+y^2-2gx-2hy+g^2+ | vedclass

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(B) The equation of the circle is $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$. Let the midpoint of a chord be $P(x_1, x_1)$ on the line $y=x$. The equation of the

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$4g^2+4h^2-8gh+4hc-4gc-c^2 < 0$

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(B) The equation of the circle is $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$. Let the midpoint of a chord be $P(x_1, x_1)$ on the line $y=x$. The equation of the chord with midpoint $(x_1, y_1)$ is $T=S_1$. Substituting $(x_1, x_1)$ for $(x_1, y_1)$,we get: $xx_1 + yx_1 - g(x+x_1) - h(y+x_1) + g^2+h^2-c^2 = x_1^2+x_1^2-2gx_1-2hx_1+g^2+h^2-c^2$. Simplifying,$x(x_1-g) + y(x_1-h) = 2x_1^2 - gx_1 - hx_1$. Since the chord passes through $(g, h+c)$: $g(x_1-g) + (h+c)(x_1-h) = 2x_1^2 - gx_1 - hx_1$. $gx_1 - g^2 + hx_1 - h^2 + cx_1 - ch = 2x_1^2 - gx_1 - hx_1$. $2x_1^2 - (2g+2h+c)x_1 + (g^2+h^2+ch) = 0$. Since there are two distinct chords,the discriminant $D > 0$: $(2g+2h+c)^2 - 8(g^2+h^2+ch) > 0$. $4g^2+4h^2+c^2+8gh+4gc+4hc - 8g^2 - 8h^2 - 8hc > 0$. $-4g^2-4h^2+8gh-4hc+4gc+c^2 > 0$. Multiplying by $-1$,we get $4g^2+4h^2-8gh+4hc-4gc-c^2 < 0$.

Two chords of the circle $x^2+y^2-2gx-2hy+g^2+h^2-c^2=0$ pass through the point $(g, h+c)$,and the line $y=x$ bisects these two chords. Then:

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