The equation of the tangent to the circle \(\mathrm{x}^{2}+\mathrm{y}^{2}+4 \mathrm{x}-4 \mathrm{y}+4=0\). Which makes equal intercepts on the positive coordinate axes is ......... (a) \(x+y=8\) (b) \(x+y=4\) (c) \(x+y=2 \sqrt{2}\) (d) \(\mathrm{x}+\mathrm{y}=2\)

When dealing with equations of circles in coordinate geometry, one method that proves highly useful is completing the square. This technique allows us to transform a quadratic equation into a form that reveals the center and radius of a circle when the equation represents one.

To complete the square, we aim to turn expressions like \(ax^2 + bx + c\) into a perfect square trinomial \(a(x-h)^2+k\), where \(h\) and \(k\) are constants that will give us the vertex of a parabola or, in the context of a circle, the center of the circle. This is done by finding a value that, when added and subtracted (or just added in the case of an equation set to zero), completes the square, hence the name of the method.

For example, in the equation \(x^2 + 4x + y^2 - 4y + 4 = 0\), we add and subtract 4 after the \(x\)-terms and add and subtract 4 after the \(y\)-terms to obtain

• \[ (x^2 + 4x + 4) + (y^2 - 4y + 4) - 4 - 4 + 4 = 0 \]
• Resulting in \( (x + 2)^2 + (y - 2)^2 = 2^2 \).

Following this method allows students to rewrite the equation of a circle in standard form and thus to identify the qualities of the circle such as radius and center more easily.

Understanding the properties of circles is essential when solving coordinate geometry problems involving circles. One key property is that the distance from the center to any point on the circle is constant and known as the radius. When a line meets the circle at only one point, it's called a tangent line. Notably, the tangent line is perpendicular to the radius of the circle at the point of contact.

The implication of this is crucial: If a tangent line creates equal intercepts on the coordinate axes, then the distances from the center to these intercepts are equal. This property informs us that the center of the circle must lie on the line bisecting the right angle formed by the intercepts, which is fundamental in finding the equation of the tangent line.

Furthermore, combining this property with the Pythagorean Theorem allows us to solve for the intercepts mathematically, as shown in our step by step solution. The principle that the square of the radius is equal to the sum of the squares of the intercepts is an application of this theorem in establishing relationships within the circle's geometry.

The subject of coordinate geometry, also known as analytic geometry, involves the study of geometric figures through the use of coordinates and the principles of algebra and analysis. It's a powerful way to analyze the position of points, lines, and shapes in the two-dimensional plane.

In the case of a circle, once its equation is known, we can use the principles of coordinate geometry to find various features such as center, radius, as well as equations of tangents and normals. For instance, understanding how the slope of a line interacts with the circle — noting that the slope of a radius and tangent are negative reciprocals if the radius is not vertical — can help when determining if a line is indeed tangent to the circle.

Moreover, coordinate geometry helps us to establish the nature of intersections, lengths of segments, areas, and other geometric concepts within the context of the Cartesian plane. It is a critical tool not only in mathematics but also in various applications in science and engineering where spatial relationships are key.