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# Tangent Circle Formula

A tangent of a circle in geometry is defined as a straight line that touches the circle at only one point. A tangent never enters the circle’s interior.

The tangent has two important properties:

- A tangent touches a circle at exactly one point on it.
- The tangent touches the circle’s radius at a right angle.

## What Is Tangent Circle Formula?

Let us now learn about the equation of the tangent. Tangent is a line and to write the equation of a line we need two things:

1. Slope (m)

2. A point on the line

General equation of the tangent to a circle:

1) The tangent to a circle equation x^{2 }+ y^{2 }= a^{2 } for a line y = mx +c is given by the equation y = mx ± a √[1+ m^{2}].

2) The tangent to a circle equation x^{2}+ y^{2 }= a^{2 }at (\(a_1, b_1)\) is x\(a_1\)+y\(b _1\)= a^{2}

Thus, the equation of the tangent can be given as xa_{1}+yb_{1 }= a^{2}, where (\(a_1, b_1)\) are the coordinates from which the tangent is made.

Let us now have a look at a few solved examples using the tangent circle formula.

## Examples using Tangent Circle Formula

**Example 1: ****Point (1,5) lies on a curve given by y=f(x)=x ^{3}-x+5. Find the equation of the tangent line to the curve that passes through the given point.**

**Solution: **To write the equation of a line we need two things:

1. Slope

2. A point on the line

It is given that the curve contains a point (1,5)

The slope is the same as the slope of the curve at x=1 which is equal to the function’s derivative at that point:

f(x) = x^{3}- x + 5

f'(x) = 3x^{2} - 1

f'(x) = 3(1)^{2} - 1 = 2

Substituting the slope m in the point-slope form of the line we would have:

\[\begin{align*} y-y_0 &= m_{tangent}(x-x_0)\\ y-5 &=2(x-1)\\ \end{align*}\]

Converting the above equation into \(y\)-intercept form as:

y-5 = 2(x-1)

y-5 = 2x-2

=2x+3

**Answer: ** The equation for the tangent line is y=2x+ 3

**Example 3: Find the equation to the pair of tangents drawn from the origin to the circle x ^{2 }+ y^{2 }- 4x - 4y + 7 = 0**

**Solution: **We use the relation obtained in the last example, T^{2} = S\({S_1},\) to write the desired equation. Here, \(({x_1},{y_1})\) is (0, 0) while g = - 2,f = - 2 and c = 7. Thus the joint equation is

\[\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&{T^2}(0,0) = S(x,y)S(0,0)\\\\ \Rightarrow & {( - 2x - 2y + 7)^2} = ({x^2} + {y^2} - 4x - 4y + 7)(7)\\\\ \Rightarrow & 4{x^2} + 4{y^2} + 49 + 8xy - 28x - 28y\\\\ &= 7{x^2} + 7{y^2} - 28x - 28y + 49\\\\ \Rightarrow & 3{x^2} - 8xy + 3{y^2} = 0\end{array}\]

**Answer: As expected, since the tangents have been drawn from the origin, the obtained equation is a homogenous one.**

## FAQs on Tangent Circle Formula

### What Is Tangent of the Circle Formula?

A tangent of a circle in geometry is defined as a straight line that touches the circle at only one point. The tangent formula is the tangent to circle equation which is y = mx ± a √[1+ m2], if the tangent is represented in the slope form and the tangent to the circle equation is x\(a_1\)+y\(b_1\)= a^{2} when tangent is given in the two-point form.

### What Is the Tangent Circle Formula?

General equation of the tangent to a circle:

1) The tangent to a circle equation x^{2} + y^{2 }= a^{2 }for a line y = mx +c is given by the equation y = mx ± a √[1+ m^{2}].

2) The tangent to a circle equation x^{2}+ y^{2} = a^{2} at (\(a_1, b_1)\) is x\(a_1\)+y\(b_1)\)= a^{2}

Thus, the equation of the tangent can be given as xa1+yb1 = a2, where (\(a_1, b_1)\) are the coordinates from which the tangent is made.

### What Is the Equation of Tangent of Circle in Slope Form?

Equation of the tangent of slope 'm', to the circle x^{2} + y^{2 }+ 2gx + 2fy + c = 0 is given by (y + f) = m(x + g) ± r √[1+ m^{2}, where r is the radius of the circle.

### What Is the Equation of the Condition of Tangency?

A line will touch a circle when the distance of the center of the circle to the line is equal to the radius of the circle, i.e., if d = r and on squaring we obtain r^{2}·(m^{2} + 1) = c^{2} the condition for a line y = mx + c to be a tangent to the circle x^{2} + y^{2} = r^{2}.

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