Types Of Differential Equations

To find the common chord, we solve the two circle equations simultaneously by subtracting them: Since the common chord is the diameter of circle , it passes through the midpoint of the segment connecting the centers of the original circles. The centers of the circles are: The midpoint of these centers, which is also the center of circle , is calculated as: However, the midpoint calculation does not match the y-value of the common chord. Adjusting to align the center of with the y-coordinate of the chord gives: This is the center of circle .

The eccentricity of an ellipse is given by the formula: where is the focal length (the distance from the center to the focus) and is the semi-major axis length (the distance from the center to the vertices along the major axis). Given that the eccentricity is , we have: The distance from the focus to the directrix is given by the equation for the focus-directrix property of the ellipse: Substituting the value of : Now, we calculate the perpendicular distance from the focus to the directrix . Using the formula for the perpendicular distance from a point to a line : For the directrix , , , and , and the point is . Substituting into the distance formula: Thus, the distance from the focus to the directrix is , which confirms that the semi-major axis length . Finally, the length of the major axis is , so: Thus, the length of the major axis is .

- Substituting points into the general equation gives a system of linear equations in terms of , , , and .
- Solve these equations to express and in terms of each other.
- Use Cramer's rule or matrix methods to find and .
- Calculate from the solved values.

Step 1: Find the slope of the tangent at the point .
- The given ellipse equation is: - Differentiate both sides implicitly with respect to : - Substitute the coordinates into the equation:

Step 2: Find the slope of the normal.
- The slope of the normal is the negative reciprocal of the slope of the tangent:

Step 3: Find the equation of the normal.
- The normal passes through the point and has a slope of . Using the point-slope form of the equation of a line:

Step 4: Substitute the equation of the normal into the ellipse equation.
- The equation of the ellipse is: - Substitute into this equation: - Expand :

Step 5: Solve the quadratic equation.
- Solve using the quadratic formula: - This gives two solutions for :

Step 6: Find the corresponding -coordinates.
- For , , which corresponds to the original point . - For , substitute into :

Step 7: Find and calculate .
- The new point of intersection is , so . - Calculate : Thus, .

The points of intersection of the given circles satisfy both equations simultaneously. These points also lie on the circle . We then use the condition that the line is tangent to circle to find its equation. First, simplify the system of the two circles by subtraction: This line and should be tangent to , implying that their intersection is the point of tangency. We now have two linear conditions in the terms of and . Using these relations, substitute into the tangent condition: The coordinates must satisfy the equation of circle , which we find by ensuring the tangent line's slope matches the derivative at this point, leading to the correct form of the circle's equation after ensuring it passes through the tangency and satisfies the tangent line equation.

We are given that the vertex of the parabola is at and the directrix is . The standard equation of a parabola with vertex and focus at (since the axis is vertical) is: where is the focal length. The equation of the directrix is , which can be rewritten as . To find the focal length , we calculate the perpendicular distance from the vertex to the directrix using the distance formula for a point to a line: where the equation of the directrix is and . For the line , , , and . Thus: The focal length is half of this distance, so: Now, the focus is at , and the intersection of the axis of the parabola (vertical line passing through the vertex) with the directrix is at . Finally, to calculate : Thus, the final answer is .