Question

The equation of circle touching the axis of \(\mathrm{y}\) at a distance \(+4\) from the origin and cutoff an intercept 6 from the axis of \(\mathrm{x}\) is \(\ldots \ldots \ldots\) (a) \(x^{2}+y^{2}-10 x-8 y+16=0\) (b) \(x^{2}+y^{2}+10 x-8 y+16=0\) (c) \(x^{2}+y^{2}-10 x+8 y+16=0\) (d) none of these

Short answer

The equation of the circle touching the axis of y at a distance +4 from the origin and cutting off an intercept 6 from the axis of x is \(x^2 + y^2 - 12x - 8y + 36 = 0\), which is not among the given options, so the correct answer is (d) none of these.

Step-by-step solution

Find the center of the circle

Since the circle touches the y-axis at a distance of +4 from the origin, the center of the circle must be at a point \((a, b)\) where \(b = 4\). Also, since it cuts off an intercept of 6 units from the x-axis, the radius, r, of the circle must be such that the distance between the center and the x-axis is half of the intercept, which is 3.

Determine the radius of the circle

Since the circle touches the y-axis at \(y = 4\), the radius of the circle must be equal to the distance between the x-axis and the center of the circle, which is 4 units. Therefore, the radius, r, is 4.

Determine the x-coordinate of the center

Now that we know the radius is 4 and the center of the circle has a y-coordinate of 4, we can use the distance between the center \((a, 4)\) and the x-axis to determine the x-coordinate of the center. Since the distance between the center of the circle and the x-axis is half of the x-intercept (3 units), we can say that the x-coordinate of the center is \(a = 3\cdot2 = 6\).

Write the equation of the circle

Now that we have the center \((6, 4)\) and the radius (4) of the circle, we can write its equation: \((x-6)^2 + (y-4)^2 = 4^2\) Expanding and simplifying the equation, we get: \(x^2 - 12x + 36 + y^2 - 8y + 16 = 16\) Combining the terms and moving constants to the right side: \(x^2 + y^2 - 12x - 8y + 36 = 0\) The correct equation for the circle is not among the given options, so the answer is: (d) none of these

Key concepts

Circle Geometry

Understanding the geometry of a circle is crucial in solving problems related to its equation. A circle is a collection of points in a plane that are equidistant from a fixed point, called the center. This fixed distance from the center to any point on the circle is known as the radius.
A circle also has a circumference, which is the total distance around the circle, and an area, which is the space contained within its circumference. In geometry, a circle can interact with other geometrical elements in various ways, such as touching axes or passing through points.
In this exercise, the problem involves a circle touching the y-axis and cutting an intercept on the x-axis, which introduces specific conditions for determining the center and radius.

Coordinate Geometry

In coordinate geometry, we use a coordinate plane to describe the location of points, lines, and shapes, like circles. The position of any point in the plane is given by an ordered pair \(x, y\), where 'x' is the horizontal component and 'y' is the vertical component.
The equation of a circle in a coordinate plane is traditionally given as: \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle, and 'r' is its radius.
Any transformation, such as translation (shifting the center) or scaling (changing the radius), affects its position and size. Solving problems related to circles in coordinate geometry involves understanding these equations and how they outline the shape's position in a graph.

Intercept on Axes

The intercepts of a circle on the coordinate axes are crucial in identifying its position and size. An intercept refers to the point where a circle or any curve intersects an axis. There are primarily two types:

• x-intercept: Points where the circle crosses the x-axis.
• y-intercept: Points where the circle crosses the y-axis.

In this exercise, we consider a circle that cuts off a segment from the x-axis, meaning it intersects the x-axis over a certain distance. If the problem specifies this intercept, it influences the radius and the center's coordinates. Understanding how intercepts work is important in deriving or confirming the circle's equation.

Circle Center

The center of a circle defines its location in the coordinate plane. It is denoted by a pair of coordinates \( (h, k) \). Knowing the center is vital because:

• It determines the horizontal (along the x-axis) and vertical (along the y-axis) shifts of the circle from the origin.
• The center directly affects the circle's equation, influencing both variables involved in the circle's representation.

In our exercise, the circle's center is influenced by its contact with the y-axis at a specified distance. This condition helps determine one of the center's coordinates, allowing us to calculate its precise location relative to the axes.

Radius Calculation

The radius of a circle is the constant distance from its center to any point on the circle. Calculating the radius involves understanding its geometric and algebraic properties within the coordinate system.
There are several ways to determine the radius:

• If the circle's equation is known, the radius can be directly obtained from the equation \( (x-h)^2 + (y-k)^2 = r^2 \), where 'r' is the radius.
• If the circle's geometrical properties, such as touching an axis or given intercepts, are specified, these can be used to derive the radius as in the exercise.

In the given problem, understanding both the y-axis touch and x-axis intercept helps ascertain the radius through logical reasoning and algebraic manipulation.