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Tangent Circle Formula determines the equations for the formation of a tangent along the circle without entering its interior. The tangent of a circle is simply defined in a branch of mathematics, that is, geometry, as a straight line touching the circle at any point. The formation of this straight line involves multiple techniques and methods of construction in geometry. In this article, we will read about tangents, circles, tangent formulas, and look at some related sample questions.
Read More:- Tangent to a Circle
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Keyterms: Tangent, Circle, Interior, Geometry, straight line, Construction, Tangency, Line
What is the Tangent Circle Formula?
The tangent to a circle can be drawn by applying a simple equation which is mentioned below.
xa1 + yb1 = a2
Here,
(a1, b1) refers to the coordinates from where the tangents are to be drawn.
What is Tangent to a Circle?
The tangent to a circle is simply defined as a straight line that touches the circle at any one or more points. However, the tangent shall not enter any circle to be correctly formed. The point at which the tangent touches the circle is known as the point of tangency.
Tangent to Circle Theorem
The tangent to a circle theorem is a defining theorem for the formation of all the tangents to a circle. Therefore, it specifiesthat any straight line shall be considered to be a tangent to a circle if and only if the line lies perpendicular to the radius drawn to the point of tangency in a circle.
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Properties of a Tangent
- A tangent is correctly formed only if touches the circle at only one point.
- A tangent never passes through a circle, that is, it never crosses the circle while entering its interior.
- A tangent is also not known for intersecting the circle at the two different points.
- As per the property of the tangent, the two tangents drawn from the single external point shall lie equal to each other, as shown in the diagram below where tangents PR and PQ are equal to each other.
Properties of a Tangent
- As per a theorem, the tangent always lies perpendicular to the radius of the circle, as shown in the diagram below where radius OP is perpendicular to the tangent RS.
Properties of a Tangent
Formation of Tangents to two circles
Suppose we have two different circles which have the same tangents or it could be said that both the non-intersecting circles share the same tangents at their different points of tangency. Now, this illustration shall be proven via given possible figures:
- In the case of the two external tangents, the possible figure shall be
- In the case of the two internal tangents, the possible figure shall be
- Now, another possible formation is seen when the two circles do not intersect but simply touch each other at just one point. In this case, there are three possible tangent lines that remain common to both as shown below.
- Another formation involves the non – intersecting but the two intrinsic circles touching each other at one point. In this case, there is just one line that is a tangent to both.
- There are also certain cases when two or more circles overlap each other and touch at two points. In such cases, there are two tangents that are common to both
- In the case of the completely concentric layers of the circles, no tangents are formed.
Things to Remember
- A tangent is a straight line that never enters the circle, however, always touches the circle.
- Tangents might be used to draw a relation between the two complementary circles.
- Contrary to the tangents are the extended chords which are also found in the straight line but it always crosses the circle at two different points, known as secant.
- A tangent can be formed in the overlapping circles or any type of two different circles only if the two different circles touch each other at any point.
- The general formula for the formation of the tangent to circle can be xa1 + yb1 = a2
Sample Questions
Ques. Given is that there is a curve line that has points (1,5) lying on it. These points are given as y = f(x) = x3 – x + 5. Find out the equation of the tangent line to the curve that passes through the given point. (4 marks)
Ans. For the equation, we require:
- Slope
- A point on the line
Given = curve line containing a point (1, 5)
The slope shall be found as
f(x) = x3 - x + 5
f'(x) = 3x2 - 1
f'(x) = 3 (1)2 - 1 = 2
Now, substituting the slope m in the point-slope form of the line.
Y − y0 = m tangent (x−x0)
Y – 5 = 2 (x − 1)
On Converting the above equation into y - intercept form
Y - 5 = 2 (x - 1)
Y - 5 = 2x -2
= 2x + 3
Hence, the equation for the tangent line shall be y = 2x + 3
Ques. What shall be the equation to the pair of tangents which are drawn from the origin to the circle x2 + y2 - 4x - 4y + 7 = 0 (4 marks)
Ans. Here, (x1, y1) (x1, y1) is (0, 0)
while g = - 2,
f = - 2 and
c = 7.
Thus, the joint equation is
(−2x −2y + 7)2 = (x2 + y2 − 4x − 4y + 7) (7)
⇒ 4x2 + 4y2 + 49 + 8xy − 28x − 28y
= 7x2 + 7y2 − 28x − 28y + 49
⇒ 3x2 − 8xy + 3y2 = 0
Therefore, the obtained equation is a homogenous one.
Ques. Define tangent to a circle with an equation. (3 marks)
Ans. The tangent to a circle is simply defined as a straight line that touches the circle at any one or more points. However, the tangent shall not enter any circle to be correctly formed.
The point at which the tangent touches the circle is known as the point of tangency.
The tangent to a circle can be drawn by applying a simple equation which is mentioned below.
xa1 + yb1 = a2
Here,
(a1, b1) refers to the coordinates from where the tangents are to be drawn.
Ques. What is the theorem of a tangent to a circle? (2 marks)
Ans. The tangent to a circle theorem is a defining theorem for the formation of all the tangents to a circle. Therefore, it specifiesthat any straight line shall be considered to be a tangent to a circle if and only if the line lies perpendicular to the radius drawn to the point of tangency in a circle.
Ques. Which properties define the tangents to a circle? (3 marks)
Ans.
- The following are the properties defining tangents to a circle.
- A tangent is correctly formed only if touches the circle at only one point.
- A tangent never passes through a circle, that is, it never crosses the circle while entering its interior.
- A tangent is also not known for intersecting the circle at the two different points.
- As per the property of the tangent, the two tangents drawn from the single external point shall lie equal to each other, as shown in the diagram below where tangents PR and PQ are equal to each other.
Ques. What can be the length of the tangent in the circle given below? (3 marks)
Ans. Given: PQ = 10 cm and QR = 18 cm,
Therefore, PR = PQ + QR = (10 + 18) cm
= 28 cm.
⇒ SR2 = PR * RQ
⇒ SR2 = 28 * 18
⇒ SR2 = 504 cm
⇒ √SR2 = √504
⇒ SR = 22. 4 cm
So, the length of the tangent is 22. 4 cm.
Ques. Find the length of the tangent in the following diagram, when the following are given -AC = 6 m and CB = 10 m. (5 marks)
Ans. The radius of a circle is perpendicular to the tangent,
triangle ABC is a right triangle, where angle A = 90 degrees.
Therefore, by the Pythagorean theorem
⇒ AB2 + AC2 = CB2
⇒ AB2 + 62 = 102
⇒ AB2 + 36 = 100
Subtract 36 on both sides.
⇒ AB2 = 100 – 36
⇒ AB2 = 64
√AB2 = √64
AB = 8.
Therefore, the length of the tangent is 8 meters.
Ques. Given is DC = 20 inch and BC = 12 inch, calculate the radius. (3 marks)
Ans. DC2 = AC * BC
But AC = AB + BC = r + 12
202 = 12 (r + 12)
400 = 12r +144
Subtract 144 on both sides.
256 = 12r
Divide both sides by 12 to get
r = 21.3
So, the radius of the circle is 21.3 inches.
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