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 the equation of the circles passing through the point

(2, 8)

and touching the lines

4 x - 3 y - 24 = 0

and

4 x + 3 y - 42 = 0

and having

x

-coordinate of the centre of the circle less than or equal to

8

.

Accepted Answer

Let $C (a, b)$ be the centre of the circle.

Given circle passes through the point $(2, 8)$ .

$∴$ Radius $(r) = \sqrt{{(a - 2)}^{2} + {(b - 8)}^{2}}$

Given circle touches the lines $4 x - 3 y = 24$ and $4 x + 3 y - 42 = 0$ .

$∴$ Perpendicular from centre to these lines is equal to the radius.

$\Rightarrow |\frac{4 a - 3 b - 24}{\sqrt{4^{2} + 3^{2}}}| = |\frac{4 a + 3 b - 42}{\sqrt{4^{2} + 3^{2}}}| = \sqrt{{(a - 2)}^{2} + {(b - 8)}^{2}} . . . (1)$

On solving first equality: $4 a - 3 b - 24 = \pm (4 a + 3 b - 42)$

Case $1 :$ Positive sign

$\Rightarrow 4 a - 3 b - 24 = 4 a + 3 b - 42$

$\Rightarrow b = 3$

Case $2 :$ Negative sign

$\Rightarrow 4 a - 3 b - 24 = - (4 a + 3 b - 42)$

$\Rightarrow a = \frac{33}{4}$

Given $|a| \leq 8 \Rightarrow - 8 \leq a \leq 8$

$∴ a \neq \frac{33}{4}$

Hence, the case $2$ assumption is wrong.

On solving $2^{n d}$ equality of equation $(1)$

$\Rightarrow |\frac{4 a + 3 b - 42}{5}| = \sqrt{{(a - 2)}^{2} + {(b - 8)}^{2}}$

Put $b = 3$ and squaring both sides

$\Rightarrow {(4 a - 33)}^{2} = (25) + {(a - 2)}^{2}$

$\Rightarrow 9 a^{2} + 164 a - 364 = 0$

Using quadratic formula:

$\Rightarrow a = \frac{- 164 \pm \sqrt{{(164)}^{2} - (36) (- 364)}}{2 \times 9}$

$\Rightarrow \frac{- 164 \pm \sqrt{40000}}{18}$

$\Rightarrow 2, \frac{- 182}{9}$

Given $|a| \leq 8$ $\Rightarrow a = 2$

$∴ a = 2 and b = 3$

$\Rightarrow r = \sqrt{{(a - 2)}^{2} + {(b - 8)}^{2}}$

$\Rightarrow r = \sqrt{25} = 5$

$∴$ Equation of required circle is ${(x - 2)}^{2} + {(y - 3)}^{2} = 25$ .

Find the equation of the circles passing through the point $(2, 8)$ and touching the lines $4 x - 3 y - 24 = 0$ and $4 x + 3 y - 42 = 0$ and having $x$ -coordinate of the centre of the circle less than or equal to $8$ .

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of the circles passing through the point 2,8 and touching the lines 4x-3y-24=0 and 4x+3y-42=0 and having x-coordinate of the centre of the circle less than or equal to 8.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of the circles passing through the point $(2, 8)$ and touching the lines $4 x - 3 y - 24 = 0$ and $4 x + 3 y - 42 = 0$ and having $x$ -coordinate of the centre of the circle less than or equal to $8$ .