Common Tangents In Coordinate Geometry: Everything You Need to Know

Common Tangents in Coordinate geometry are an important concept of higher mathematics. A tangent itself holds a lot of value in competitive exams like JEE and NEET. When you ride a bicycle then, the straight path on which the cycle moves acts as a tangent for the tyre of your bicycle.

Table of Contents

What is Common Tangent in coordinate geometry?
Important Facts
Direct common tangents
Transverse common tangents
When do two circles intersect orthogonally
Practice Problems
FAQs

What is the Common Tangent in Coordinate Geometry?

A common tangent in coordinate geometry is a straight line that touches two or more circles at different locations. At the points of contact, this tangent line has the same slope as the radii of the circles. Common tangents in coordinate geometry are frequently employed in situations involving numerous circles and serve a vital role in establishing the link between them.

Important Facts

It is crucial to comprehend the following points before learning about the many kinds of frequent tangents:

Both internal and exterior tangents are common. Outside of the circles are external common tangents, while inside the circles are internal common tangents.
Depending on how close together two circles are, they may have zero, one, or two shared tangents.
The length of all tangents taken from a central point outside the circles is the same.

Direct Common Tangents

Tangents that cross the line connecting the centres of two circles are known as direct common tangents in coordinate geometry. These diversions fall into one of three categories:

a) Case 1: No Intersection - The circles do not cross, and there are no direct shared tangents if the separation between their centres is larger than the total of their radii.

b) Case 2: One Tangent - The circles touch externally and have a single straight common tangent when the distance between their centres equals the sum of their radii.

c) Case 3: Two Tangents - The circles meet at two different locations if the distance between their centres is smaller than the product of their radii. There are two straight common tangents in coordinate geometry in this situation.

Transverse Common Tangents

Tangents known as transverse common tangents do not cross the line connecting the centres of the circles. Transverse common tangents in coordinate geometry can also be categorised into three situations, like direct common tangents:

a) Case 1: No Intersection- There are no transverse common tangents in coordinate geometry if the distance between the centres of the circles is larger than the total of their radii.

b) Case 2: One Tangent- The circles contact internally and have a single transverse common tangent when the separation between their centres matches the difference between their radii.

c) Case 3: Two Tangents- The circles are distinct and have two transverse common tangents in coordinate geometry if the distance between their centres is smaller than the difference in their radii.

When Do Two Circles Intersect Orthogonally?

Two circles that cross at right angles are said to be orthogonal. The following circumstance makes this phenomenon possible:

The slopes of the tangents derived from the intersection of two circles are added together, and the result is a negative one.
This requirement guarantees that the tangents are parallel to one another, leading to an orthogonal intersection.

Practice Problems

Q1. What can be said about two circles' shared tangents if their radii are 12 units in total and their centres are 10 units apart?

A) There are no common tangents.
B) There is one direct common tangent.
C) There are two direct common tangents.
D) There is one transverse common tangent.

Answer: (C) Two

Explanation: The distance between the centres of the two circles is greater than the sum of their radii. Therefore, only two tangents can be drawn.

Q2. If the distance between two circles' centres is greater than the total of their radii, how many common tangents can there be between them?

A) Zero
B) One
C) Two
D) Cannot be determined

Answer: (C)Two

Explanation: The radii of the two circles added together do not equal the distance between their centres. There may be two direct common tangents in this situation. This is so that the circles can touch the outside since the centres are spaced far enough away.

Q3. What may be said about the common tangents if the distance between the centres of two circles is larger than the total of their radii?

A) There are no common tangents.
B) There is one direct common tangent.
C) There are two direct common tangents.
D) There is one transverse common tangent.

Answer: (C) There are two direct common tangents.

Explanation: Similar to the preceding query, there can be two straight common tangents if the separation between the centres of the two circles is greater than the sum of their radii.

FAQs

Q1. Can circles have more than two common tangents?
Answer. No, circles may only share a maximum of two tangents.

Q2. Can circles intersect at more than two points?
Answer. No, circles can only come together at two specific locations.

Q3. How can I determine if two circles touch externally or internally?
Answer. To establish if two circles contact internally or externally, compare the distance between their centres with the sum or difference of their radii.