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Let's go through each of the six exercises and discuss the steps required to solve them:

1. **Tangents to a Circle at the Points on the X-axis**  
   The equation of the circle is \( x^2 + y^2 - 8x + 10y - 128 = 0 \).  
   To find the tangent lines to this circle at points on the x-axis, follow these steps:
   - Find the points of intersection of the circle with the x-axis (\(y = 0\)).
   - At these points, calculate the slope of the tangent lines.
   - Use the point-slope form of a line to derive the equations of the tangents.

2. **Equation of a Circle Tangent to a Line**  
   The given line is \( 3x + 2y + 4 = 0 \), and the point is \((-2,1)\), with the center on \( x - 8y + 36 = 0 \).
   - Write the general form of the equation of the circle: \( (x - h)^2 + (y - k)^2 = r^2 \).
   - Use the conditions of tangency, distance from the center to the line, and the point condition to find the center \((h, k)\) and radius \(r\).
   - Use these to write the equation of the circle.

3. **Equation of Circle Through A(3,5) Tangent to Line**  
   Given points are \( A(3,5) \) and \( B(-1,1) \), and the line of tangency is \( 3x + y + 2 = 0 \).
   - Use the general form \( (x - h)^2 + (y - k)^2 = r^2 \).
   - Utilize the conditions for the points passing through the circle and the tangent at point B.
   - Solve for \(h\), \(k\), and \(r\) accordingly.

4. **Circle Containing Three Given Points**  
   The points are \( A(1,1), B(-2,3), C(3,4) \).  
   - You need to find the circle that passes through these three points.
   - Use the general equation \( x^2 + y^2 + Dx + Ey + F = 0 \), and substitute the coordinates of the points to form a system of equations.
   - Solve this system to determine the values of \(D\), \(E\), and \(F\), and thus the equation of the circle.

5. **Line Containing Diameter of a Circle Parallel to Given Line**  
   The equation of the circle is \( x^2 + y^2 + 2x - 4y - 4 = 0 \), and the line must be parallel to \( 3x + 5y = 4 \).
   - Find the center and radius of the circle by completing the square.
   - Identify a line that passes through the center and contains a diameter. Since it is parallel to \( 3x + 5y = 4 \), its slope will be the same as that of the given line.

6. **Prove Diagonals of a Rhombus are Perpendicular**  
   Use coordinate geometry to prove that the diagonals of a rhombus are perpendicular:
   - Place the vertices of the rhombus at convenient coordinates on the plane.
   - Write the equations of the diagonals and show that their slopes are negative reciprocals of each other, confirming perpendicularity.

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Would you like detailed steps on any specific problem? Here are five related questions to deepen your understanding:

1. How do you derive the equation of a tangent to a circle at any point on its circumference?
2. How can you determine the distance from a point to a line in coordinate geometry?
3. What conditions must a line satisfy to be tangent to a circle at a given point?
4. How do you prove that a line is parallel to another using slope concepts?
5. What is the geometric significance of the perpendicularity of diagonals in a rhombus?

**Tip:** Always double-check tangency conditions by calculating the perpendicular distance from the center of the circle to the tangent line!

Find the equations (in general form) of the tangent lines to the circle x^2 + y^2 - 8x + 10y - 128 = 0 at the points of the circle on the x-axis. Find the equation (in general form) of the circle tangent to the line 3x + 2y + 4 = 0 at (-2, 1) and whose center is on the line x - 8y + 36 = 0. Find an equation of the circle containing the point A(3,5) and tangent to the line 3x + y + 2 = 0 at B(-1,1). Write an equation of the circle (in general form) containing the points A(1,1), B(-2,3), & C(3,4). Find an equation of the line that contains a diameter of the circle x^2 + y^2 + 2x - 4y - 4 = 0, and which is parallel to the line 3x + 5y = 4. Prove analytically (using coordinate geometry) that the diagonals of a rhombus are perpendicular.

Learn how to find the equations of tangents to a circle, determine circle equations given tangency conditions, and prove geometric properties using coordinate geometry. This guide covers essential geometry concepts, including circle equations, tangents, and properties of rhombuses. Suitable for students in Grades 10-12.

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