Problem 72
Question
Suppose that \((a, b)\) is on the circle \(x^{2}+y^{2}=r^{2}\). Show that the line \(a x+b y=r^{2}\) is tangent to the circle at \((a, b)\).
Step-by-Step Solution
Verified Answer
The line \(ax + by = r^2\) is tangent to the circle at \((a, b)\) because they have exactly one point of intersection.
1Step 1: Understanding the Problem
We need to show that the line given by the equation \(ax + by = r^2\) is tangent to the circle \(x^2 + y^2 = r^2\) at the point \((a, b)\). A line is tangent to a circle at a given point if it intersects the circle at exactly one point, that point being the point of tangency.
2Step 2: Substitute the Point into Circle Equation
Since \((a, b)\) is on the circle \(x^2 + y^2 = r^2\), substituting \(x = a\) and \(y = b\) into the circle equation confirms that \(a^2 + b^2 = r^2\). This indicates that the point \((a, b)\) satisfies the circle equation and is on the circle.
3Step 3: Verify Line Intersection
Substitute \(x = a\) and \(y = b\) into the line equation \(ax + by = r^2\): \(aa + bb = r^2\) which simplifies to \(a^2 + b^2 = r^2\). Since \(a^2 + b^2 = r^2\) is already known from the circle equation, this confirms that \((a, b)\) also satisfies the line equation.
4Step 4: Check for Tangency
To ensure tangentness, we need to confirm that no other point besides \((a, b)\) lies on both the circle and the line. Since both equations equate to \(r^2 = a^2 + b^2\), and \(ax + by = r^2\) equates to the same expression when \((x, y) = (a, b)\), \((a, b)\) must be the only intersection point. Hence, the line is tangent to the circle at \((a, b)\).
Key Concepts
Circle EquationPoint of TangencyIntersection PointsCircle Geometry
Circle Equation
A circle is a perfectly round shape on a plane, and its equation is a mathematical representation of all the points that make up the circle. The common form of a circle's equation is given by \(x^2 + y^2 = r^2\), where \r\ is the radius of the circle, and the center is at the origin, point \(0, 0\).
The equation means that for any point \(x, y\) on the circle, the sum of the squares of its coordinates is equal to the square of the radius. This ensures that all points are the same distance from the center.
In our exercise, you can see the direct application of this equation to verify if a point lies on the circle, which is critical in proving that a line is tangent to the circle.
The equation means that for any point \(x, y\) on the circle, the sum of the squares of its coordinates is equal to the square of the radius. This ensures that all points are the same distance from the center.
In our exercise, you can see the direct application of this equation to verify if a point lies on the circle, which is critical in proving that a line is tangent to the circle.
Point of Tangency
The point of tangency is where a line just touches a circle and does not cross it. At this point, the line is perpendicular to the radius of the circle drawn to the point.
For a line to be tangent at \(a, b\) on \(x^2 + y^2 = r^2\), it needs to intersect the circle at precisely one point, which happens to be \(a, b\) in this scenario.
For a line to be tangent at \(a, b\) on \(x^2 + y^2 = r^2\), it needs to intersect the circle at precisely one point, which happens to be \(a, b\) in this scenario.
- This single point of intersection indicates that they meet at the boundary of the circle only once.
- The condition for tangency requires that the equation of the line equates directly with the circle's equation when the coordinates are \(a, b\).
Intersection Points
Intersection points occur whenever a given line crosses a curve, like a circle. If there is exactly one intersection point between a line and a circle, the line is tangent to the circle, indicating a very specific kind of intersection.
In this exercise, we observe the intersection between the line \(a x + b y = r^2\) and the circle \(x^2 + y^2 = r^2\), which meets at \(a, b\).
In this exercise, we observe the intersection between the line \(a x + b y = r^2\) and the circle \(x^2 + y^2 = r^2\), which meets at \(a, b\).
- The process involves confirming that substituting \(a, b\) into both the circle and line equations yields a true statement, specifically \(r^2\).
- If substituting other points into the equations does not satisfy them, then \(a, b\) is indeed the only intersection point.
Circle Geometry
Circle geometry deals with the properties and relations of points on a circle. Essential concepts include radius, diameter, tangent lines, and chords, among others.
In the context of this exercise, understanding the properties of tangent lines—specifically their uniqueness and perpendicular relation to the radius—is crucial.
In the context of this exercise, understanding the properties of tangent lines—specifically their uniqueness and perpendicular relation to the radius—is crucial.
- A tangent line at any given point \(a, b\) on a circle will only touch the circle at that point.
- The geometric idea reinforces that this line forms a right angle with the radius at \(a, b\).
Other exercises in this chapter
Problem 71
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