Orthogonal Circles: Definition, Conditions & Diagrams Explained
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Orthogonal Circles are two circles intersecting at right angles. If two circles intersect in two points, and the radii drawn to the points of intersection meet at right angles, then the circles are orthogonal, and the circles can be said to be perpendicular to each other. Thus they are also known as perpendicular circles.
In this maths article, we will learn about Orthogonal Circles, the Orthogonality Theorem, the Condition of Orthogonality of Circles, Perpendicular Circles, How to Draw a pair of Orthogonal Circles with Diagrams.
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What is a Circle?
A circle is a curved line that runs around a centre point. It is a basic topic of geometry. Every part of the curved line is the same distance from the centre. A circle has a circumference and it can be described by the equation of a circle. In other words, it is the locus of all points that are equidistant from the origin. In the two-dimensional plane, the circle is the shape enclosing the most area per unit perimeter squared.
What are Orthogonal Circles?
When two circles cut orthogonally they are orthogonal curves and called orthogonal circles of each other. A circle orthogonal to another circle means the angle between two circles is equal to 90. When this condition is satisfied then the circles are said to be orthogonal. Hence, they are also called perpendicular circles. Orthogonal circles are invariant under inversion. Application of orthogonal circles includes Problem of Plane Strain in Plasticity and much more.
Know how to calculate the Area of a circle in the linked article.
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Condition of Orthogonality of Circles
Two curves are said to be orthogonal if their angle of intersection is a right angle i.e the tangents at their point of intersection are perpendicular. Orthogonal trajectories of the family of circles are sets of circles having the same condition of orthogonality.
Circles intersecting orthogonally are orthogonal curves. By the Pythagorean theorem, the equation of orthogonal circles with two circles of radii r_1 and r_2 whose centres are a distance d apart are orthogonal if
Two circles with Cartesian equations
The condition for orthogonality is
For orthogonal circles S = 0 and S’ = 0, a tangent of S = 0 at the point of intersection will be normal to S’ = 0.
Hence it passes through the centre of S’ = 0. And vice-versa.
It is important to note that,
- S = 0 and S’ = 0 are touching circles, S – S’ = 0 or S’ – S = 0 is a common tangent.
- If S = 0 and S’ = 0 are intersecting circles, S – S’ = 0 or S’ – S = 0 is a common chord of the circle.
Also, read about Arithmetic progressions with this article.
If two circles α and β are orthogonal, then
- The tangents at each point of intersection pass through the centres of the other circle.
- Each circle is its own inverse with respect to the other.
These conditions are important to solve questions on orthogonal circles like finding the condition for two circles to be orthogonal.
In the picture above, the two circles intersect orthogonally at points A and B. The point P’ is the inverse of P with respect to the blue circle. As the point P moves around the black circle, the inverse P’ also moves around the black circle.
You can move the points A and B to resize the black circle and move the points O and X to resize the blue circle.
Learn about various parts of a circle here!
How to Draw a pair of Orthogonal Circles
Follow these simple steps to draw orthogonal circles perfectly.
- Draw the first radius of any size you want.
- Draw a tangent to the circle at a particular point. If you need the tangent at a specific angle from the horizontal axis use a protractor to plot the point of tangency.
- Draw a perpendicular line to the tangent passing through the centre of the circle.
- Draw the second circle of the desired radius of the circle with its centre on the tangent of the first circle.
To check the orthogonality of the circle’s measures angle ATO.
- Similarly, 3 orthogonal circles can be drawn and the equation of circle cutting orthogonally the three circles can be found.
Learn the various concepts of the Binomial Theorem here.
Orthogonality Theorem
If the circles
Proof:
Let m_1 and m_2 be the slopes of the circles respectively.
From the properties we know that,
Differentiating
Therefore,
Differentiating
Since both curves intersect at point (u,v) the point should satisfy the curves. Hence, we replace x and y by u and v.
u+v=1
Orthogonal Circles Solved Examples
Q1. Prove that the two circles will cut orthogonally if 2gg' + 2ff' = c + c'
A1. \(\begin{array}{l}\text{Let } 0(-\mathrm{g},-\mathrm{f}) \& 0^{\prime}\left(-\mathrm{g}^{\prime},-\mathrm{f}^{\prime}\right) \text{ be the centres of the circles of radius } \mathrm{r} \text{ and } \mathrm{r}^{\prime}\text{ respectively.}\\
\text{As the circles cut orthogonally there radii are perpendicular to each other,thus}\\
\Rightarrow \mathrm{r}^{2}+\mathrm{r}^{\prime 2}=\left(00^{\prime}\right)^{2} \\
\Rightarrow \mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}+\mathrm{g}^{\prime 2}+\mathrm{f}^{\prime 2}-\mathrm{c}^{\prime}=\left(\mathrm{g}-\mathrm{g}^{\prime}\right)^{2}+\left(\mathrm{f}-\mathrm{f}^{\prime}\right)^{2} \\
\Rightarrow \mathrm{g}^{2}+\mathrm{f}^{2}-\mathrm{c}+\mathrm{g}^{2}+\mathrm{f}^{\prime 2}-\mathrm{c}^{\prime}=\mathrm{g}^{2}+\mathrm{g}_{1}^{2}-2 \mathrm{gg}^{\prime}+\mathrm{f}^{2}+\mathrm{f}_{1}^{2}-2 \mathrm{ff}^{\prime} \\
\Rightarrow 2 \mathrm{gg}^{\prime}+2 \mathrm{ff} \mathrm{f}^{\prime}=\mathrm{c}+\mathrm{c}^{\prime}\\
\text{Hence, proved.}\end{array}\)
Q2. Show that the circles
A2. From
Condition to prove two circles are orthogonal:
Thus,
Hence proved.
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FAQs For Orthogonal Circles
How do you know if two circles are orthogonal?
If two circles intersect in two points, and the radii drawn to the points of intersection meet at right angles, then the circles are orthogonal.
What is the condition for circles to intersect orthogonally?
Two curves are said to be orthogonal if their angle of intersection is a right angle i.e the tangents at their point of intersection are perpendicular. Two circles with Cartesian equations
How do you draw orthogonal circles?
Follow these simple steps to draw orthogonal circles perfectly. Draw the first radius of any size you want. Draw a tangent to the circle at a particular point. If you need the tangent at a specific angle from the horizontal axis use a protractor to plot the point of tangency. Draw a perpendicular line to the tangent passing through the centre of the circle. Draw the second circle of the desired radius with its centre on the tangent of the first circle.
What is the orthogonality theorem?
If the circles
What is a Circle?
A circle is a curved line that runs around a centre point. Every part of the curved line is the same distance from the centre. In other words, it is the locus of all points that have equidistant from the origin. In the two-dimensional plane, the circle is the shape enclosing the most area per unit perimeter squared.
What does the term “orthogonal” mean in geometry?
“Orthogonal” means “perpendicular.” When two curves or lines meet at a right angle (90°), they are said to be orthogonal.
Do orthogonal circles always intersect?
Yes, for two circles to be orthogonal, they must intersect at two points, and at each point, the angle between their tangents must be 90°.
Can a circle be orthogonal to itself?
No, a circle cannot be orthogonal to itself. Orthogonality requires two different circles.
Is the condition for orthogonality dependent on the radius?
Not directly. The condition depends on the center coordinates and constant terms in the circle equations, which are related to radius but not solely dependent on it.
Can a straight line be orthogonal to a circle?
Yes, a line can be orthogonal (perpendicular) to a circle at the point where it meets the radius. However, this is different from two circles being orthogonal.