Equation of Pair of Tangents to a Circle

Here you will learn what is the equation of pair of tangents to a circle from an external point with example.

Let’s begin –

Equation of Pair of Tangents to a Circle

Let the equation of circle S = \(x^2\) + \(y^2\) – \(a^2\) and P(\(x_1,y_1\)) is any point outside the circle. From the point we can draw two real and distinct tangent and combine equation of pair of tangents is –

(\(x^2\) + \(y^2\) – \(a^2\))(\({x_1}^2\) + \({y_1}^2\) – \(a^2\)) = \(({xx_1 + yy_1 – a^2})^2\)

or  \(SS_1 = T^2\)

where \(S_1\) = \({x_1}^2\) + \({y_1}^2\) – \(a^2\)

and \(T^2\) = \(({xx_1 + yy_1 – a^2})^2\)

Also Read : Length of Tangent to a Circle Formula From an External Point

Example : Find the equation to the pair of tangents drawn from (4, 10) to the circle \(x^2 + y^2\) = 4.

Solution : Given that, S = \(x^2 + y^2\) = 4 and Point P is (4, 10).

Then, by using above formula,

\(S_1\) = \(4^2 + 10^2 – 4\) = 112

\(T^2\) = \((4x + 10y – 4)^2\) =  \((4x + 10y)^2\) + 16 – 8(4x + 10y)

= \(16x^2\) + \(100y^2\) + 80xy + 16 – 32x – 80y

= \(16x^2\) + \(100y^2\) – 32x – 80y + 80xy + 16

Hence, the combined equation of pair of tangents is

\(x^2 + y^2 – 4\)(112) = \(16x^2\) + \(100y^2\) – 32x – 80y + 80xy + 16

\(96x^2\) + \(12y^2\) – 32x – 80y + 80xy – 464 = 0

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