Problem 123 - A Complete Resource Book in Mathematics for JEE Main 2017

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 [2007] (A) \(0

Step-by-Step Solution

Answer

(B) \(k \geq \frac{1}{2}\).

Step 1: Identify Circle Properties

A circle is described by the equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. For a circle to be tangent to the x-axis, the distance from the center to the x-axis must be equal to the radius. Thus, \(k = r\). Furthermore, the circle passes through the point \((-1, 1)\).

Step 2: Substitute Conditions into Circle Equation

Using \((x-h)^2 + (y-k)^2 = r^2\) and substituting the point \((-1,1)\) gives: \((-1-h)^2 + (1-k)^2 = k^2\).

Step 3: Expand and Simplify the Equation

Expand the equation: \((-1-h)^2 + (1-k)^2 = h^2 + 2h + 1 + k^2 - 2k + 1 = k^2\). Simplifying, we have: \(h^2 + 2h + 2 - 2k = 0\).

Step 4: Solve for k

Organize the equation: \(2k = h^2 + 2h + 2\). Thus, \(k = \frac{h^2 + 2h + 2}{2}\).

Step 5: Analyze Range of Values for k

Since the radius \(r = k > 0\) and the selected conditions, \(h^2 + 2h + 2 > 0\). Completing the square for the expression \(h^2 + 2h + 2\), we have \((h+1)^2 + 1\), which is always greater than or equal to \(1\). Thus, \(k \geq 1/2\).

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. In the context of circles, geometry helps us understand the relationship between different components such as radius, diameter, and the arc length. A **circle** is a two-dimensional shape where every point on the surface is equidistant from the center. This distance is known as the **radius**. The line segment joining the center to any point on the circle is always the same length, which is crucial when solving geometric problems related to circles.

• The **center** is a fixed point within the circle equidistant to all points on the circle's circumference.
• A **radius** is a line segment from the center to the circumference.
• The **diameter** is twice the radius, extending from one side of the circle to the other through the center.
• The **circumference** is the distance around the circle.

Understanding these basics of geometry helps in comprehending the behavior and formulas involving circles, such as the equation of a circle.

Tangent

A tangent to a circle is a line that touches the circle at exactly one point. This point is known as the point of tangency. Tangents are significant in geometry because they help to solve problems involving properties of circles and their line interactions. In this exercise, the circle is tangent to the x-axis, which means:

• The x-axis touches the circle at just one point.
• The radius of the circle is perpendicular to the tangent at the point of tangency.

For the circle in this exercise, the x-axis behaves as a tangent line. Therefore, if the center of the circle is \( (h, k) \), then the radius must be equal to \( k \) because the distance from the center of the circle to the x-axis is \( k \), as the y-coordinate value \( k \) is the perpendicular distance from the x-axis.

Equation of Circle

The equation of a circle in a Cartesian coordinate system is expressed as \( (x-h)^2 + (y-k)^2 = r^2 \). Here, \( (h, k) \) are the coordinates of the center, and \( r \) is the radius. This fundamental equation helps in finding various circle properties.

• If a circle is tangent to an axis, like in this problem, then the radius is equal to the distance from the center to that axis.
• Substituting specific points that the circle passes through into the equation allows us to find relationships or constraints on the variables.

In this problem, using the equation of the circle, we substitute the point \( (-1,1) \) to establish a relationship: \[ (-1-h)^2 + (1-k)^2 = k^2 \]. Solving this equation provides a valuable link between \( h \) and \( k \), leading to the realization that \( k \) is determined as \[ k = \frac{h^2 + 2h + 2}{2} \]. This relationship is crucial for determining the possible range of \( k \).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses the Cartesian coordinate system to study geometric shapes. It combines algebra and geometry, providing a bridge between these two branches of mathematics. In coordinate geometry:

• Points are defined by ordered pairs \( (x, y) \).
• This system allows for algebraic equations to be used to describe geometric figures.
• Helps in deriving properties like the distance between two points or the equation of lines and curves.

For the problem addressed in this exercise, coordinate geometry allows us to visualize the position of the circle relative to the x-axis and solve for variable coordinates efficiently. By representing the circle's center and the tangent line within this system, we can calculate precise conditions required for tangency and passing through specific points, ultimately deriving constraints on \( k \). Such analysis leads to determining the range of \( k \), which in this case, must meet \( k \geq \frac{1}{2} \).