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**6. A diameter of a circle:**
   - (a) Find an equation of the circle \( C_1 \), which has endpoints at \((-3, -5)\) and \((7, 3)\).
   - (b) Another circle \( C_2 \) has center \((10, 5)\) and the same radius as \( C_1 \). Find the equation of the circle \( C_2 \).
   - (c) The two circles intersect at points \( R \) and \( S \). Show that the equation of the line \( RS \) is \( y = -2x + 13 \).
   - (d) Show that the \( x \)-coordinates of \( R \) and \( S \) satisfy the equation \( 5x^2 - 60x + 159 = 0 \).

**7. The coordinates of two points \( A \) and \( B \) are \((-7, 3)\) and \( (5, 11) \) respectively:**
   - (a) Show that the equation of the perpendicular bisector of \( AB \) is \( 3x + 2y = 11 \).
   - (b) A circle passes through \( A \) and \( B \), and its center lies on the line \( 12x - 5y = 70 \). Find the equation of the circle.

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A diameter of a circle C1 has endpoints at (-3, -5) and (7, 3). Another circle C2 has center (10, 5) and radius same as C1. The two circles intersect at points R and S. Find an equation of the circle C1. Show that the equation of line RS is y = -2x + 13 and hence that the x-coordinates of R and S satisfy the equation 5x^2 - 60x + 159 = 0. Also, given points A (-7, 3) and B (5, 11), find the equation of the perpendicular bisector of AB, and find the equation of a circle passing through A and B, whose center lies on 12x - 5y = 70.

Learn how to solve geometry problems involving circles, lines, and simultaneous equations. This includes finding the equation of a circle, the line of intersection, and solving quadratic equations. Suitable for high school students studying advanced geometry.

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