Question

The centre of a circle is \(\mathrm{C}(2, \mathrm{k})\). If \(\mathrm{A}(2,1)\) and \(\mathrm{B}(5,2)\) are two points on its circumference, then the value of \(\mathrm{k}\) is (1) 6 (2) 2 (3) \(-6\) (4) \(-2\)

Short answer

Based on the given information and solution, determine the coordinates of the center of the circle. The coordinates of the center of the circle C are (2, k). We found that the value of k is 2. Therefore, the coordinates of the center of the circle are (2, 2).

Step-by-step solution

Write the distance formula

The distance formula between any two points P(x1, y1) and Q(x2, y2) is given by: \[\sqrt{(x2 - x1)^2 + (y2 - y1)^2}\]

Apply the distance formula for points C and A, and C and B

Apply the distance formula to find the distance between C(2,k) and A(2,1) as well as C(2,k) and B(5,2). Let r be the radius of the circle. For CA: \[r = \sqrt{(2-2)^2 + (1-k)^2}\] For CB: \[r = \sqrt{(5-2)^2 + (2-k)^2}\]

Set the radius equal for both distances

Set the distance from C to A to be equal to the distance from C to B, as both are equal to the radius of the circle. \[\sqrt{(2-2)^2 + (1-k)^2} = \sqrt{(5-2)^2 + (2-k)^2}\]

Solve for k

Square both sides of the equation to eliminate the square root and simplify: \[(2-2)^2 + (1-k)^2 = (5-2)^2 + (2-k)^2\] Solve for k: \[k^2 - 2k = 0\] Factor out k: \[k(k - 2) = 0\] So we have two possible values for k: 0 and 2. We note that k=0 would result in point C having the same coordinates as point A, which would mean that there would be no circle with distinct points. Thus, we must have k=2.

Final Answer

The value of k is \(\boxed{2}\) (Option 2).

Key concepts

Distance Formula

The distance formula is a key concept in coordinate geometry, useful for calculating the distance between two points in a plane. This formula originates from the Pythagorean theorem and is vital in problems involving circles, lines, and triangles. It is expressed as: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where

• \(x_1\) and \(y_1\) are the coordinates of the first point,
• \(x_2\) and \(y_2\) are the coordinates of the second point.

For example, if we want to find the distance between two points, such as C(2, k) and A(2, 1), you simply substitute the coordinates into the formula and simplify. This process provides a straightforward method for locating the radius of a circle or verifying geometrical properties. Understanding how to handle and apply the distance formula is an essential skill in solving various geometric problems at the Mathematics Class 10 level.

Circle Equation

In coordinate geometry, the equation of a circle is crucial when assessing circular shapes and areas in the plane. The standard form of the equation of a circle is:\[ (x - h)^2 + (y - k)^2 = r^2 \]where:

• \((h, k)\) represents the center of the circle,
• \(r\) is the radius.

This formula helps to identify the position and size of a circle given its center and circumference points. For instance, in the original problem, C(2, k) is the center, and points A(2,1) and B(5,2) lie on the circle's circumference. By recognizing that the distance from the center to any point on the circle is constant, we can determine the radius using the distance formula. The equation thereby assists in solving for unknowns such as the values of \(k\) or the radius itself, playing a pivotal role in various geometric problem-solving tasks.

Geometric Problem Solving

Geometric problem-solving entails applying various mathematical principles and formulas to understand and tackle problems involving shapes, sizes, and relative positions of figures. This type of problem-solving involves logic and spatial reasoning, often requiring steps like identifying key points, noting distances, or symmetry principles. When working on problems like the one presented, it is essential to:

• Clearly identify what is given and what needs to be found.
• Set up equations based on geometric properties, like equal radii in a circle.
• Solve systematically to find unknown values, such as variable \(k\) in the circle problem.

Utilizing a step-by-step approach not only avoids confusion but also ensures that all variables are accounted for. This logical structuring can sometimes involve rearrangement or factoring, as seen when solving for \(k\) where the equation \(k(k - 2) = 0\) led us to possible solutions by factoring.

Mathematics Class 10

Mathematics Class 10 serves as a pivotal stage in education where students encounter more complex concepts that form the foundational basis for higher-level math. Topics typically include algebra, geometry, trigonometry, and basic statistics. In geometric problem-solving, class 10 students are introduced to key coordinate geometry topics, such as the distance formula and circle equations. Understanding these topics is crucial as they not only reiterate fundamental principles but also develop critical thinking skills. The practical application of math problems, like determining the center of a circle or calculating a radius using points on the circumference, helps strengthen conceptual understanding. Through engaging with problems from textbooks and exercises, students in class 10 can enhance their mathematical fluency and prepare for advanced studies. Whether solving for variables, graphing equations, or examining properties of shapes, these exercises build the analytical groundwork essential for future success in mathematics.