Name: Tangents to a Circle From an External Point
Uploaded: 2019-07-19
Duration: 3 min 5 s
Description: Do you know the equation of the tangents to a circle drawn from an external point? Let's find the equation involving the slope, which helps to find the equation of tangents.

Question

Find an equation of the circle which touches the straight lines

x + y = 2, x - y = 2

and also touches the circle

x^{2} + y^{2} = 1

.

Accepted Answer

Clearly, lines $x + y = 2$ and $x - y = 2$ intersect at $(2, 0)$ . So, the centre of the required circle lies on the bisector of the angle containing the origin.

Let, $r (r > 0)$ be the radius of the required circle. Then, there are two possible circles

The coordinates of the centres are $C_{1} (1 + r, 0)$ and $C_{2} (- (1 + r), 0)$

And the circles touch the lines $x + y - 2 = 0 & x - y - 2 = 0$ .

Applying the condition of tangency of a line to the first circle we get, $\frac{| (1 + r) \pm 0 - 2 |}{\sqrt{2}} = r$

$\Rightarrow |r - 1)| = \sqrt{2} r$

$\Rightarrow {(r - 1)}^{2} = 2 r^{2}$

$\Rightarrow r^{2} + 2 r - 1 = 0$

$\Rightarrow r = \pm \sqrt{2} - 1$

$\Rightarrow r = \sqrt{2} - 1$ (As $r > 0$ )

Thus, coordinates of the centre and radius of the first possible circle are $(\sqrt{2}, 0)$ and $(\sqrt{2} - 1)$ , respectively.

Hence, the equation of the required circle is

$(x - \sqrt{2})^{2} + y^{2} = (\sqrt{2} - 1)^{2}$ .

Again, for another circle with centre $C_{2} (- 1 - r, 0)$ ,

Applying condition of tangency, we get,

$\frac{| - (1 + r) \pm 0 - 2 |}{\sqrt{2}} = r$

$\Rightarrow |r + 3| = \sqrt{2} r$

$\Rightarrow r (\sqrt{2} - 1) = 3$ (considering the positive value only).

$\Rightarrow r = \frac{3}{\sqrt{2} - 1} = 3 (\sqrt{2} + 1)$ .

Hence, the centre of the other circle $C_{2} (- 3 \sqrt{2} - 4, 0)$ and radius is $r = 3 \sqrt{2} + 3$ .

Find an equation of the circle which touches the straight lines $x + y = 2, x - y = 2$ and also touches the circle $x^{2} + y^{2} = 1$ .

Important Questions on Circle

Learn and Prepare for any exam you want !

Find an equation of the circle which touches the straight lines x+y=2, x-y=2 and also touches the circle x2+y2=1.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find an equation of the circle which touches the straight lines $x + y = 2, x - y = 2$ and also touches the circle $x^{2} + y^{2} = 1$ .