A circle is given by \(\mathrm{x}^{2}+(\mathrm{y}-1)^{2}=1\), another circle \(\mathrm{C}\) touches it externally and also the \(\mathrm{x}\) -axis, then the locus of its centre is \(\ldots \ldots \ldots\) (a) \(\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}=4 \mathrm{y}\right\\}\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=0\\}\) (b) \(\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}+(\mathrm{y}-1)^{2}=4\right\\}\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=0\\}\) (c) \(\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}=\mathrm{y}\right\\} \quad\\{(0, \mathrm{y}): \mathrm{y}=0\\}\) (d) \(\left\\{(x, y): x^{2}=4 y\right\\}\\{(0, y): y=0\\}\)

Understanding the equations of circles is fundamental in coordinate geometry, particularly when dealing with problems that involve finding the locus of a circle's center. A standard circle equation is expressed as \( (x-h)^{2} + (y-k)^{2} = r^{2} \), where \( (h, k) \) is the center, and \( r \) is the radius.

When given \( x^{2} + (y-1)^{2} = 1 \), we identify the center at \( (0, 1) \) and a radius of 1. Any circle equation in this format can be translated to find the center and radius, allowing us to graph it accurately and understand its relationship with other geometric figures, such as lines, points, and indeed, other circles.

Exercise Improvement Advice

In working with exercises involving circle equations, it's essential to emphasize the role of the center and radius. We can improve students' understanding by creating interactive content allowing learners to manipulate these values and observe real-time changes in the circle’s representation on the coordinate plane.

Coordinate geometry, also known as analytic geometry, is a branch of geometry that utilizes a coordinate system to investigate the relationships of geometrical shapes and figures. It enables us to algebraically solve geometric problems and to translate between algebraic equations and geometric curves.

In the case of the circle with its center on the locus, coordinate geometry principles are used to derive the locus equation. By knowing the distance between the centers of the touching circles and recognizing that this distance is the sum of their radii, we establish a geometric relationship that can be represented algebraically. Moreover, if a circle touches the x-axis, the y-coordinate of the center equals the radius - this is another instance of coordination between algebraic and geometric understanding.

Exercise Improvement Advice

To enhance comprehension in exercises on coordinate geometry, it would be beneficial to integrate visual aids that show the gradual construction of loci. Demonstrating steps that tie algebra with geometry can solidify understanding and allow students to visualize the transition from formula to figure.

External tangency is an interesting concept where two circles or a circle and another shape are touching each other at exactly one point, but do not intersect. In our problem, we are working with two circles that touch each other externally and the circle 'C' also touches the x-axis, which is a straight line.

The tangency imposes a set of conditions: where the circles touch each other, the center-to-center line must be perpendicular to the line of tangency, and the distance between the centers is exactly equal to the sum of the radii of the two circles (since they do not overlap). These properties are crucial in deriving the locus of the center of the external circle.

Exercise Improvement Advice

To aid students in grasping the concept of external tangency, interactive simulations where students can move one circle around another until they achieve tangency can be highly instructional. Additionally, highlighting the properties of tangency – such as the fact that tangent circles share a common tangent line at the point of tangency – with diagrams and live examples can offer more clarity than purely algebraic explanations.