Name: Pair of Tangent, Chord of Contact
Uploaded: 2019-07-19
Duration: 19 min 37 s
Description: This video contains all the details of the chord of the contact pair of tangent and cord bisected at a point. This concept is very useful for the students.

Question

Find the equations of the common tangents of the circles

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

and

{(x - a)}^{2} + y^{2} = b^{2}

.

Accepted Answer

Given: The circles are $x^{2} + y^{2} = c^{2} . . . (1)$ and ${(x - a)}^{2} + y^{2} = b^{2} . . . (2)$

From figure, centres and radii of the circles are $C (0, 0), C' (a, 0)$ and $r_{1} = c, r_{2} = b$ , respectively,

As we know that $\frac{C T}{C' T} = \frac{C P}{C' P} = \frac{r_{1}}{r_{2}} . . . (3)$ ,

Applying section formula we get the coordinates of

$T (\frac{c \cdot a + b \cdot 0}{c + b}, 0)$ (internal division)

$\Rightarrow T (\frac{a c}{c + b}, 0)$

and $P (\frac{c \cdot a - b \cdot 0}{c - b}, 0)$ (external division)

$\Rightarrow P (\frac{a c}{c - b}, 0)$

Now equation of common tangents(Transverse common tangents) through $T (\frac{a c}{c + b}, 0)$ is given by the equation

$(y - 0) = m (x - \frac{a c}{c + b})$

$\Rightarrow m x - y - \frac{m a c}{c + b} = 0 . . . (4)$

Similarly, the equation of common tangents (Direct Common tangents) through $P (\frac{a c}{c - b}, 0)$ is given by

$m x - y - \frac{m a c}{c - b} = 0 . . . (5)$

Applying condition of tangency for the tangent $(4)$ to the circle $(1)$ , we get

$|\frac{m (0) - (0) - \frac{m a c}{c + b}}{\sqrt{1 + m^{2}}}| = c$

$\Rightarrow {(\frac{m a c}{c + b})}^{2} = c^{2} (1 + m^{2})$ (squaring both sides)

$\Rightarrow \frac{m^{2} a^{2}}{{(c + b)}^{2}} = (1 + m^{2})$

$\Rightarrow m^{2} (\frac{a^{2}}{{(c + b)}^{2}} - 1) = 1$

$\Rightarrow m^{2} = \frac{{(c + b)}^{2}}{(a^{2} - {(c + b)}^{2})}$

$\Rightarrow m = \pm \frac{|(b + c)|}{\sqrt{(a^{2} - {(b + c)}^{2})}}$

Similarly,applying condition of tangency for the tangent $(5)$ of the circle $(1)$ we get

$m = \pm \frac{|(b - c)|}{\sqrt{(a^{2} - {(b - c)}^{2})}}$

$∴$ The equation of common tangents are $y = m x + c \sqrt{1 + m^{2}}$ , where

$m = \frac{\pm (b + c)}{\sqrt{a^{2} - (b + c)^{2}}} or \frac{\pm (b - c)}{\sqrt{a^{2} - (b - c)^{2}}}$

Find the equations of the common tangents of the circles $x^{2} + y^{2} = c^{2}$ and ${(x - a)}^{2} + y^{2} = b^{2}$ .

Important Questions on The Circle

Learn and Prepare for any exam you want !

Find the equations of the common tangents of the circles x2+y2=c2 and x-a2+y2=b2.

Important Questions on The Circle

Learn and Prepare for any exam you want !

Find the equations of the common tangents of the circles $x^{2} + y^{2} = c^{2}$ and ${(x - a)}^{2} + y^{2} = b^{2}$ .