Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

The locus of centre of the circle which touches the circle \(x^{2}+(y-1)^{2}=1\) externally and also touches \(x\)-axis is (A) \(\left\\{(x, y): x^{2}+(y-1)^{2}=4\right\\} \cup\\{(x, y): y<0\\}\) (B) \(\left\\{(x, y): x^{2}=4 y\right\\} \cup\\{(0, y): y<0\\}\) (C) \(\left\\{(x, y): x^{2}=y\right\\} \cup\\{(0, y): y<0\\}\) (D) \(\left\\{(x, y): x^{2}=4 y\right\\} \cup\\{(x, y) ; y<0\\}\)

Step-by-Step Solution

Answer

Option (D): \(\{(x, y): x^2 = 4y\} \cup \{(x, y) ; y < 0\}\) is the answer.

Step 1: Identify Center and Radius

The given circle equation is \(x^2 + (y-1)^2 = 1\). Comparing with the standard form \((x-h)^2 + (y-k)^2 = r^2\), we find the center is \((0, 1)\) and radius \(r = 1\).

Step 2: Condition for Circle that Touches Given Circle and X-axis

The circle that touches the given circle externally and the x-axis must have a center \((x, y)\) such that the distance from \((x, y)\) to \((0, 1)\) is equal to the sum of their radii, and the distance from \((x, y)\) to the x-axis must be equal to its radius.

Step 3: Apply External Tangency Condition

For the two circles to touch externally, the distance between their centers must equal the sum of their radii. Thus, \(\sqrt{x^2 + (y - 1)^2} = 1 + r\), where \(r\) is the radius of the circle centered at \((x, y)\).

Step 4: Apply Tangency Condition With X-axis

The circle also touches the x-axis, so the vertical distance from \((x, y)\) to the x-axis must equal its radius \(r\). Therefore, \(y = r\).

Step 5: Solve for the Locus

From Step 3, we have \(\sqrt{x^2 + (y - 1)^2} = 1 + y\). From Step 4, \(y = r\). Substitute \(y\) for \(r\) in the tangency equation to get \(\sqrt{x^2 + (y-1)^2} = 1 + y\). Simplifying, \(x^2 + (y-1)^2 = (1+y)^2\). Expanding and simplifying gives \(x^2 = 4y\).

Step 6: Consider the Additional Condition y<0

Since a circle cannot have a negative radius, the part of \(x^2 = 4y\) where \(y < 0\) is not relevant to the original circle conditions but can be seen as an additional locus condition when considering non-standard circles.

Key Concepts

Circle Tangency

Understanding circle tangency is essential in solving geometry problems involving circles interacting with each other, or with other lines. When we say two circles touch each other externally, it means they meet at exactly one point and their centers are on opposite sides of the touching point.
This is different from internally tangent circles, where one circle lies inside the other and they share a common tangent.
In problems like these, clarity around the term 'tangency' is crucial, as it affects the geometric configuration of the problem. The point of tangency is unique where the circles or lines are tangent.

External Tangency

External Tangency occurs when the distance between the centers of two circles equals the sum of their radii.
This relationship helps define the geometric configuration between two tangent circles.
Understanding external tangency aids in setting up equations to find the precise condition to satisfy this relation.
For two circles that touch externally:

• The distance between their centers is the sum of their radii: if one has radius 1 at point (0,1), and the other meets at (x,y) and radius r, then \ \({\sqrt{x^2 + (y - 1)^2} = 1 + r}\).
• This forms a critical equation when determining the locus of a moving circle which adheres to these constraints.

Analytic Geometry

Analytic Geometry is the bridge that connects algebra and geometry, providing tools to solve geometric problems using algebraic equations.
In this context, it allows us to derive the potential path or locus of a circle's center that is externally tangent to another circle and a line, like the x-axis.
In problems with external tangency and touching lines, it gives us the ability to set conditions such as the equality in distances from a point to a line or between two centers.
For example, we utilize the equation of a circle, \ \((x - h)^2 + (y - k)^2 = r^2}\), to find relationships between coordinates like \ \(x^2 = 4y\), which reflect parabola properties, defining a boundary or locus.

Circle Geometry

Circle Geometry involves studying properties, relationships, and movements of circles.
When applied in problems involving tangency, it focuses on how circles intersect, touch, or are tangent to lines and other circles.
This is exemplified when finding loci of centers that satisfy given conditions, such as contact points with other geometric entities.

• The locus of centers is given by geometric transformations and conditions of tangency, resulting in a new figure, commonly a parabola or a line.
• This helps visualize and define the path which a circle's center can take while maintaining its tangency conditions.
• An additional, sometimes overlooked, condition in circles geometry is extending solutions beyond usual constraints, such as recognizing potential loci in negative ranges due to symmetry or algebraic derivation.