Name: Two Circles Touching Internally and Externally
Uploaded: 2019-07-19
Duration: 6 min 6 s
Description: In this video, we will learn the condition for two circles touching internally and externally using an interesting example. Come, let's learn and watch this...

Question

Find the pair tangents drawn from

(1, 3)

to the circle

x^{2} + y^{2} - 2 x + 4 y - 11 = 0

and also find the angle between them.

Accepted Answer

The given equation of the circle is $S = x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ and the point from which a pair of tangents is drawn is $P (1, 3)$ .

Here, the centre of the given circle is at $(1, - 2) and radius r = \sqrt{1 + 4 + 11} = 4$

The equation of the pair of tangents from a point $(x_{1}, y_{1})$ to the circle $S = 0$ is given by $S_{11} S = S_{1}^{2}$

$\Rightarrow (1^{2} + 3^{2} - 2 \cdot 1 + 4 \cdot 3 - 11) (x^{2} + y^{2} - 2 x + 4 y - 11) = {\{x \cdot 1 + y \cdot 3 - 1 (x + 1) + 2 (y + 3) - 11\}}^{2}$

$\Rightarrow 9 x^{2} + 9 y^{2} - 18 x + 36 y - 99 = {(5 y - 6)}^{2}$

$\Rightarrow 9 x^{2} + 9 y^{2} - 18 x + 36 y - 99 = 25 y^{2} - 60 y + 36$

$\Rightarrow 9 x^{2} - 16 y^{2} - 18 x + 96 y - 135 = 0$

As we know that if $θ$ is the angle between the tangents through a point $(x_{1}, y_{1})$ to a circle $S = 0$ then $\tan (\frac{θ}{2}) = \frac{r}{\sqrt{S_{11}}}$

So, $\tan (\frac{θ}{2}) = \frac{4}{\sqrt{9}} = \frac{4}{3}$

By using trigonometric identity of $\sin 2 x = \frac{2 \tan x}{1 + \tan^{2} x}$

$\Rightarrow \sin θ = \frac{2 \tan \frac{θ}{2}}{1 + \tan^{2} \frac{θ}{2}} = \frac{\frac{8}{3}}{1 + \frac{16}{9}}$

$\Rightarrow \sin θ = \frac{24}{25} \Rightarrow \cos θ = \sqrt{1 - \frac{576}{625}}$

$\Rightarrow \cos θ = \sqrt{\frac{49}{625}} = \frac{7}{25}$

$∴ θ = \cos^{- 1} (\frac{7}{25})$

Thus, the equation of the pair of tangents from $(1, 3)$ to the circle $x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ is $9 x^{2} - 16 y^{2} - 18 x + 96 y - 135 = 0$ and the angle between them $\cos^{- 1} (\frac{7}{25})$ .

Find the pair tangents drawn from $(1, 3)$ to the circle $x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ and also find the angle between them.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the pair tangents drawn from 1,3 to the circle x2+y2-2x+4y-11=0 and also find the angle between them.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the pair tangents drawn from $(1, 3)$ to the circle $x^{2} + y^{2} - 2 x + 4 y - 11 = 0$ and also find the angle between them.