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# What is the equation of a tangent to a circle from an external point?

**Solution:**

A tangent is a straight line that touches the circle at a point and is called the tangent to the circle. The point where the tangent touches the circle is called the point of contact.

Tangent to a circle is the line drawn from an outside point or the point on the circumference of the circle, that touches the circle at a unique single point on its circumference. Tangent is always perpendicular to the radius of the circle. Point of tangency refers to the point at which the tangent meets the circle.

Line AB in the above diagram is the tangent to the circle, with point C as the point of tangency.

Various conditions of tangency:

Tangent is supposed to be made from a point onto the circle. Thus, there are three basic conditions of study for a tangent drawn to the circle from a point. They are:

1) When the point lies on the circle itself :

In this case, only one tangent can be constructed.

Case 2- When the point lies outside the circle

In this case, exactly 2 tangents are possible.

Case 3: When the point lies inside the circle

In this case, no tangent is possible, since a tangent is a line that touches the circumference of the circle but never intersects it.

General equation of the tangent to a circle:

1) The tangent to a circle with 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 with 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 x\(a_1\)+y\(b_1\)_{ }= a^{2}, where \((a_1, b_1)\) are the coordinates of the external point from which the tangent is drawn.

## What is the equation of a tangent to a circle from an external point?

**Summary:**

The equation of the tangent can be given as x\(a_1\)+y\(b_1\)_{ }= a^{2}, where (\(a_1\), \(b_1\)) are the coordinates from which the tangent is drawn.

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