Tangent Line In A Circle

Tangent is based on a Latin discussion meaning "to bear on". A tangent is a direct line or a plane that touches the circumference of any surface at any ane indicate. For case, one real-world example of a tangent is when y'all ride a bicycle, every point on the circumference of the wheel makes a tangent with the road. In this article, nosotros will larn what a tangent to a circle is and how to construct a tangent to a circle. We will study the equation of a tangent to a circumvolve, the length of a tangent to a circle from an external indicate and the equation of a tangent to a circumvolve in slope form. Also, nosotros will learn some theorems on tangents to circles and atmospheric condition with the help of bug.

In this mathematics commodity, we are going to learn nigh what is tangent and how to draw tangents to a circle ina a very easy fashion. Theorems and equations based on a tangent to a circle, solve some problems on a tangent to a circle that will aid you to understand the topic hands.

Tangent to a Circle

A straight line touching, only not intersecting a circle is what we call a tangent to a circle. In other words, a tangent line to a circle is a line that touches the circumvolve at exactly ane point.

Tangent to a Circle

Hither, in the to a higher place circle with centre O, straight line PL is a tangent which touches the circumvolve at point P. It touches the circumvolve at only one point and looks similar a line that sits but outside the circle'south circumference. There are some important points related to the tangent to a circle which are as follows:

At that place is but ane unique tangent at a given point of the circle.

The tangent of a circumvolve is a special case of the secant when two end points of its corresponding chord coincide.

Tangent to a Circle

From the above figure, we notice that the secant line PQ becomes a tangent line as Q approaches P along the circumference of a circumvolve or points P and Q coincide.

A tangent line is ever perpendicular to the radius of the circle drawn from the heart to the point where the tangent touches the circle.

Equation of Tangent to a Circle

Consider a circle with eye O(a,b) and a radius of '\(r\)' units as shown in the figure in which P(x,y) is a point on the circumference of the circle. The equation of this circumvolve is :

\(\left ( ten-a \right )^{2} + \left ( y-b \correct )^{two}=r^{ii}\)

Tangent to a Circle

From the figure, we see that tangent line AB touches the circle at point P.

We know, the radius of the circle is perpendicular to the tangent at the point of contact P,

OP\(\perp\) AB,

and \(\measuredangle OPA = \measuredangle OPB = 90^{0}\)

Too, we know that the product of the slope of the radius and that of the tangent line = -1.

i.e. \(m_{OP}\times m_{AB}\) = -1

At present, we decide the equation of a tangent line to a circle:

Step 1: Firstly find the equation of the circle and write it in the form,

\(\left ( ten-a \right )^{ii} + \left ( y-b \right )^{two}=r^{2}\)

Pace two: From the in a higher place equation, find the coordinates of the middle of the circle (a,b)

Step three: Discover the gradient of the radius –

\(m_{OP} = \frac{y_{2} – y_{1}}{x_{2} – x_{ane}}\)

Step 4: Since the radius is perpendicular to the tangent of the circle at a betoken P,

\(m_{AB} = -\frac{1}{m_{OP}}\)

Stride v: Now write down the slope point form for the equation of the tangent line AB and substitute the value found for \(m_{AB}\) and the coordinates of P. This will give the states the equation of the tangent line.

\(y – y_{1} = m(x – x_{ane})\)

Construction of a tangent to a circumvolve

Follow the beneath steps to construct the tangent to a circle:

Footstep ane: First depict a circumvolve using a compass with the given radius and centre O.

Tangent to a Circle

Pace 2: Now, draw a line that will join the centre of the circle and whatever point P on the circumference of the circle. Here, OP is the radius of the circle.

Tangent to a Circle

Pace 3: Set the compass point on P and set it to any width less than the radius OP. So, on the line simply drawn (i.due east. OP), draw an arc on each side of P. This gives the points Q and R as shown in the figure.

Tangent to a Circle

Stride four: Fix the compass on Q and fix information technology to any width greater than the distance QP. Without changing the compass width, draw an arc approximately in the position shown in the figure on i side of P.

Tangent to a Circle

Step 5: At present without irresolute the width of the compass, motility the compass to point R and make another arc across the commencement arc, mark this point S.

Tangent to a Circle

Step 6: Now join the points P and S.

Tangent to a Circle

Stride vii: We get the line PS which is the tangent to a circle at point P.

Theorem on Tangent to a Circumvolve

Theorem 1:

The tangent at any point of a circumvolve is perpendicular to the radius through the point of contact.

Theorem 2:

The lengths of the two tangents from an external point to a circle are equal. Consider the following diagram:

Hither, Air-conditioning=BC.

Condition of Tangency

A tangent is considered when it touches the surface of a circle at merely one point and that bespeak where information technology touches the circle's surface is called the indicate of tangency. Thus, based on the signal of tangency and where it is situated w.r.t the circle, we have defined the below conditions for tangent as follows:

When Point Lies Outside the Circle:

From the given effigy, we run across that

Circle

At that place are exactly 2 tangents to a circle from a point which lies outside the circle.

When Point Lies Within the Circumvolve:

From the given effigy, we meet that

Circle

No tangent can be fatigued to a circle which passes through a point lying within the circle.

When Indicate Lies on the Circle:

From the given figure we see that

Circle

At that place is only one tangent to a circle through a point which lies on the circumvolve

Length of Tangent to a Circle From an External Point

The length of tangent from the external betoken P(\(x_{1},y_{ane}\)) to the circumvolve

\(ten^{2} + y^{2} + 2gx + 2fy + c = 0\) is

\(\sqrt{}\left ( x_{ane}^{2}+y_{1}^{2}+2gx_{i}+2fy_{1}+c \right )=\sqrt{S_{1}}\)

External Point

Here in the given figure, the tangent to a circle is PT and the external betoken is P.

Notation:

To notice the length of the tangent. Let S = \(10^{2} + y^{2} + 2gx + 2fy + c = 0\) so, \(S_{ane} = x_{ane}^{ii}+y_{1}^{2}+2gx_{1}+2fy_{ane}+c\). Where, P(\(x_{ane},y_{1}\)). Therefore, length of tangent = \(\sqrt{S_{1}}\)

For \(S_{i}\) first write the equation of circle in general class i.e. coefficient of \(x^{2}\) = coefficient of \(y^{2}\) = 1 and making the R.H.Due south. of circle is zip, then let L.H.S by S.

Properties of Tangent

A tangent is a direct line touching, just non intersecting a circle.
A tangent is perpendicular to the radius of the circle at the point of tangency.
A tangent tin can never intersect the circumvolve at two points, information technology just touches the circle at one point.
The length of tangents from an external betoken to a circle is always equal.

Tangent to a Circle Solved Examples

ane. Detect the length of tangents drawn from the point (iii, -four) to the circle \(2x^{2} + 2y^{2} – 7x – 9y – xiii = 0\).

Solution: The equation of the given circle is

\(2x^{2} + 2y^{2} – 7x – 9y – 13 = 0\)

Rewrite the given equation of the circle equally

i.e. \(x^{2} + y^{two} – \frac{vii}{2}x – \frac{9}{two}y – \frac{13}{two} = 0\)

Allow S = \(x^{two} + y^{2} – \frac{seven}{ii}x – \frac{9}{2}y – \frac{thirteen}{two} = 0\)

Therefore, \(S_{1}=\left(3\correct)^{2}+\left(-4\right)^{2}-\frac{seven}{ii}\times3-\frac{9}{2}\times\left(-four\right)-\frac{13}{2}\)

On solving the to a higher place expression, we get

\(S_{ane}=26\)

Therefore, Length of tangent = \(\sqrt{S_{ane}} = \sqrt{26}\)

2. Find the equations of tangents to the circle \(10^{2}+y^{ii}=16\) drawn from the point (i, 4).

Solution: Equation of line through (1, 4) is (y – four) = m(x – 1)

\(\Rightarrow\) mx – y + 4 – thousand = 0 …………(i)

So, perpendicular length from centre(0, 0) to mx – y + four – m = 0 is equal to radius

and so, \(\frac{\left|iv-1000 \right|}{\sqrt{\left ( thou^{ii}+1 \right )}}=4\)

Or \(\left(4-grand\right)^{2} = 16\left(grand^{2}+1\right)\)

\(\Rightarrow\) \(15m^{two}+8m=0\)

\(\Rightarrow\) \(m\) = 0,\(\frac{-8}{15}\)

From equation(i), the tangents from (i, 4) are y = four and 8x + 15y = 68.

We hope that the above article is helpful for your understanding and exam preparations. Stay tuned to the Testbook App for more updates on related topics from Mathematics, and various such subjects. Also, accomplish out to the test series available to examine your noesis regarding several exams.

Tangent to a Circle FAQs

Q.1 How many tangents can exist drawn to a circle?

Ans.i Infinite. We know that for any circumvolve we can draw a tangent at whatsoever betoken on the circumference and this would mean we can depict infinitely many tangents for a circle.

Q.2 How many parallel tangents tin exist drawn to a circumvolve?

Ans.2 Two parallel lines. A circle can have at most 2 parallel tangents, i at a indicate on information technology and the other at a point diametrically opposite to it.

Q.three How to observe the length of tangent to a circle?

Ans.3 Suppose '\(r\)' be the radius of circle, '\(d\)' be the distance from heart of circumvolve to the external bespeak and '\(t\)' is the length of tangent, and so \(t\) = \(\sqrt{\left ( d^{two}-r^{ii} \right )}\)

Q.4 Are tangents of a circle equal?

Ans.4 Tangents drawn to a circle from the same point outside the circumvolve are equal in length.

Q.v How many tangents tin can be drawn to two circles touching externally?

Ans.v Three common tangents can be drawn passing through two circles touching externally.

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