Problem 123 - A Complete Resource Book in Mathematics for JEE Main

Question

Consider a family of circles which are passing through the point \((-1,1)\) and are tangent to \(x\)-axis. If \((h, k)\) are the co-ordinates of the centre of the circles, then the set of values of \(k\) is given by the interval (A) \(0

Step-by-Step Solution

Answer

The correct interval for \(k\) is \(k \geq \frac{1}{2}\) (Option B).

Step 1: Identify Key Properties of the Circle

For a circle tangent to the x-axis, the distance from its center to the x-axis must be equal to the radius. Therefore, if the center is at \((h, k)\), then the radius \(r = k\) because the circle is tangent to the x-axis at \(y = 0\).

Step 2: Write the Equation of the Circle

The general equation of a circle with center at \((h, k)\) and radius \(r\) is \(x - h)^2 + (y - k)^2 = r^2\). Since \(r = k\), this becomes \((x - h)^2 + (y - k)^2 = k^2\).

Step 3: Substitute Point \\((-1, 1)\\) into the Equation

Since the circle passes through the point \((-1, 1)\), substitute \(x = -1\) and \(y = 1\) into the equation. We have:\((-1 - h)^2 + (1 - k)^2 = k^2\).

Step 4: Simplify the Equation

Expanding and simplifying the equation, we get: \( (h + 1)^2 + (1 - k)^2 = k^2 \). Simplify further by expanding the squared terms:\(h^2 + 2h + 1 + 1 - 2k + k^2 = k^2\).

Step 5: Eliminate the \\(k^2\\) Term and Rearrange

Subtract \(k^2\) from both sides to get:\(h^2 + 2h + 2 - 2k = 0\).

Step 6: Determin Values for \\(k\\) for a Real Solution

To solve for \(k\), rearrange the equation to\(-2k = -h^2 - 2h - 2\).Thus, \(k = \frac{h^2 + 2h + 2}{2}\).

Step 7: Identify the Range of \\(k\\)

Considering the complete equation for varying values of \(h\) would give:\(k = \frac{h^2 + 2h + 2}{2}\), and since \(k\) must be positive for the circle to be above the x-axis, simplify to observe: Minimum value of \(h^2 + 2h + 2 = 2\), hence \(k \geq \frac{1}{2}\). Therefore, the set of values for \(k\) is when \(k\) is greater than or equal to \(\frac{1}{2}\) and not specifically restricted otherwise.

Key Concepts

Circles

A circle is a collection of all points in a plane that are equidistant from a given point, known as the center. This distance is called the radius. Understanding circles is fundamental in coordinate geometry as they form the basis for various geometrical concepts and real-life applications.
In the coordinate plane, a circle can be represented mathematically by an equation that encapsulates the coordinates of its center and its radius. This simple equation is key to exploring many properties and behaviors of circles, from size and shape to its interactions with other geometric figures.
Circles can intersect with lines in different ways. They can be tangent to lines, pass through them, or be completely separate. For a circle tangent to a line, such as the x-axis, the distance from the center to this axis equals the radius. Hence, understanding the nature of tangent lines plays a crucial role in solving problems involving circles.

Tangency Conditions

The term 'tangency' refers to a point where a circle touches a line, a condition requiring specific geometry. For the circle to be tangent to the x-axis, its center lies directly above the x-axis and the vertical distance from the center to the axis is exactly its radius.
Mathematically, if you have a circle centered at \(h, k\) with radius \(r\), and it is tangent to the x-axis, it holds that \(r = k\). This equation shows that for the circle to be tangent to the x-axis, the distance from the circle's center to the axis (along the y-axis) must be equal to the radius.

• This tangency condition directly influences the equation of the circle.
• It also plays a critical role when solving problems involving tangent circles.

Understanding tangency helps us predict and define the behavior of circles in a coordinate plane, and reinforces the spatial relationship circles have with lines.

Equations of Circles

The equation of a circle in a coordinate plane reveals much about its size and position. Given a center \(h, k\) and radius \(r\), the circle's equation takes on a familiar form: \((x - h)^2 + (y - k)^2 = r^2\).
This form illustrates a few important points:

• The values \(h\) and \(k\) represent the horizontal and vertical translations of the circle's center from the origin.
• \(r^2\) determines the area of the circle, with a larger radius yielding a larger area.

For circles passing through a specific point, like \((-1,1)\), substitute these coordinates into the circle's equation to derive further constraints.
Solving such equations requires methodically substituting values and simplifying, often revealing specific conditions like those needed for tangency. These mathematical manipulations allow assessment of geometric properties, such as the suitable intervals for \({k\) when the circle is constrained to touch the x-axis and pass through a given point.