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(D) The equation of the common chord (line $L$) of two circles ${S_1} = 0$ and ${S_2} = 0$ is given by ${S_1} - {S_2} = 0$.Given circles are:${S_1}: {

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$0$

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(D) The equation of the common chord (line $L$) of two circles ${S_1} = 0$ and ${S_2} = 0$ is given by ${S_1} - {S_2} = 0$.Given circles are:${S_1}: {x^2} + {y^2} - 25 = 0$${S_2}: {x^2} + {y^2} - 8x + 7 = 0$Subtracting the two equations:$({x^2} + {y^2} - 25) - ({x^2} + {y^2} - 8x + 7) = 0$$8x - 32 = 0$$x = 4$This is the equation of the line $L$.The second circle is ${x^2} + {y^2} - 8x + 7 = 0$. Comparing this with the general form ${x^2} + {y^2} + 2gx + 2fy + c = 0$,we get $2g = -8$,so $g = -4$,and $f = 0$.The centre of the second circle is $(-g, -f) = (4, 0)$.The length of the perpendicular from the point $(4, 0)$ to the line $x - 4 = 0$ is given by:$d = \frac{|4 - 4|}{\sqrt{1^2 + 0^2}} = \frac{0}{1} = 0$.Thus,the length of the perpendicular is $0$.

The line $L$ passes through the points of intersection of the circles ${x^2} + {y^2} = 25$ and ${x^2} + {y^2} - 8x + 7 = 0$. The length of the perpendicular from the centre of the second circle onto the line $L$ is

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