Problem 42
Question
If the tangents \(P Q\) and \(P R\) are drawn to the circle \(x^{2}+\) \(y^{2}=a^{2}\) from the point \(P\left(x_{1}, y_{1}\right)\), then the equation of the circumcircle of \(\Delta P Q R\) is (A) \(x^{2}+y^{2}-x x_{1}-y y_{1}=0\) (B) \(x^{2}+y^{2}+x x_{1}+y y_{1}=0\) (C) \(x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0\) (D) none of these
Step-by-Step Solution
Verified Answer
(A) The equation is \( x^2 + y^2 - x x_1 - y y_1 = 0 \).
1Step 1: Understand the Concept of Tangents to a Circle
The question involves tangents drawn to a circle from an external point. The tangents from point \( P(x_1, y_1) \) to the circle \( x^2 + y^2 = a^2 \) are \( P Q \) and \( P R \). The important property of a tangent is that it is perpendicular to the radius at the point of tangency.
2Step 2: Determine the Power of a Point
The power of the point \( P(x_1, y_1) \) with respect to the circle \( x^2 + y^2 = a^2 \) is given by the equation: \( x_1^2 + y_1^2 - a^2 \). This leads to the equation \( PQ = PR \) and the relationship that the length from \( P \) to the point of tangency squared is equal to the power of the point.
3Step 3: Equation of Tangents to the Circle
The equation of a tangent from point \( P(x_1, y_1) \) to the circle \( x^2 + y^2 = a^2 \) is derived using the point of contact for the tangents. The tangent equation is \( (xx_1 + yy_1) = a^2 \).
4Step 4: Derive the Equation of the Circumcircle
The general form of the equation of a circle is \( (x - x_1)^2 + (y - y_1)^2 = r^2 \). However, for circumcircle calculations involving tangents PQ and PR, the circle passes through \( P(x_1, y_1) \) and is equidistant to points \( Q \) and \( R \), we use the midpoint theorem and symmetry to examine the relationship. The equation of the circle can then be simplified using known relationships into: \( x^2 + y^2 - xx_1 - yy_1 = 0 \).
5Step 5: Compare Given Options
The derived equation from the previous step matches option (A): \( x^2 + y^2 - x x_1 - y y_1 = 0 \). This is the equation corresponding to the circumcircle of \( \Delta P Q R \) that passes through point \( P \) and ensures symmetry with respect to the tangents.
Key Concepts
Tangents to a CirclePower of a PointEquation of TangentsProperties of Tangents
Tangents to a Circle
When you have a circle and a tangent, it's like drawing a line just touching a single point on the circle's edge. Imagine touching a balloon with the tip of a pencil—it's just a point contact. This point where the tangent touches the circle is called the point of tangency.
A key fact about tangents is their relationship with the radius. The radius is the line from the center of the circle to any point on its edge. The tangent and the radius at the point of tangency are perpendicular. Think of it as an ‘L’ shape. So, if you know where the tangent line meets the circle, you can always be sure it's at a right angle to the radius.
When drawing tangents from any external point, two tangents can be drawn that are equal in length. This can be helpful when calculating distances or verifying your work in geometric problems.
A key fact about tangents is their relationship with the radius. The radius is the line from the center of the circle to any point on its edge. The tangent and the radius at the point of tangency are perpendicular. Think of it as an ‘L’ shape. So, if you know where the tangent line meets the circle, you can always be sure it's at a right angle to the radius.
When drawing tangents from any external point, two tangents can be drawn that are equal in length. This can be helpful when calculating distances or verifying your work in geometric problems.
Power of a Point
The power of a point relates to circles and external points. It's a measure of how far a point is from being on a circle. It's calculated with the expression: \(x_1^2 + y_1^2 - a^2\) for point \(P(x_1, y_1)\)and circle \(x^2 + y^2 = a^2\).
This relationship tells us something important. If we have the tangents from a point to a circle, the squared length of these tangents equals the power of the point. This concept can simplify complex geometrical configurations by providing a consistent way to relate points and circles.
Using power of a point can assist in solving problems involving multiple tangents or proving equal tangent lengths. It provides a neat mathematical tool to handle tangents systematically.
This relationship tells us something important. If we have the tangents from a point to a circle, the squared length of these tangents equals the power of the point. This concept can simplify complex geometrical configurations by providing a consistent way to relate points and circles.
Using power of a point can assist in solving problems involving multiple tangents or proving equal tangent lengths. It provides a neat mathematical tool to handle tangents systematically.
Equation of Tangents
To find the equation of a tangent from any point to a circle, you can use a specific formula. If the point is \(P(x_1, y_1)\)and the circle is given by \(x^2 + y^2 = a^2\), the equation of the tangent is: \((xx_1 + yy_1) = a^2\).
This equation acknowledges the internal symmetry of circles and uses the geometric properties to relate the tangents and the point. It reflects the linear relationship between the circle and the external point.
Exploring this equation further allows you to see how tangents behave as you move point \(P\).Whether it moves closer or farther from the circle, the equation adjusts to maintain its correctness, showing the flexibility and robustness of this algebraic form.
This equation acknowledges the internal symmetry of circles and uses the geometric properties to relate the tangents and the point. It reflects the linear relationship between the circle and the external point.
Exploring this equation further allows you to see how tangents behave as you move point \(P\).Whether it moves closer or farther from the circle, the equation adjusts to maintain its correctness, showing the flexibility and robustness of this algebraic form.
Properties of Tangents
Tangents to a circle possess several unique properties that can be very useful. One key property is that from a point outside the circle, the two tangents drawn will always be of equal lengths. This is because each tangent creates a path of shortest distance from the external point to the circle, maintaining consistency.
Additionally, at the point of tangency, tangents are perpendicular to the radius. This key relationship ensures that circles retain their perfect round shape and is also crucial in many proofs and derivations in geometry.
Tangents can also transform into cyclic quadrilaterals, where opposite angles sum up to 180 degrees. This happens when tangents are extended and intersect at external points, forming unique shapes and configurations.
Additionally, at the point of tangency, tangents are perpendicular to the radius. This key relationship ensures that circles retain their perfect round shape and is also crucial in many proofs and derivations in geometry.
Tangents can also transform into cyclic quadrilaterals, where opposite angles sum up to 180 degrees. This happens when tangents are extended and intersect at external points, forming unique shapes and configurations.
- Equal lengths from external point
- Perpendicular to radius
- Helps form special quadrilaterals
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