Find the radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are $2 x + 4 y + 3 = 0$ , $4 x + 3 y + 3 = 0$ , and $x + 1 = 0$ .

Open in App

Solution

$2 x + 4 y + 3 = 0..... (i) 4 x + 3 y + 3 = 0..... (i i) x + 1 = 0........ (i i i)$

Solving $(i i)$ and $(i i i)$ we get $A (- 1, \frac{1}{3})$

solving $(i i i)$ and $(i)$ we get $B (- 1, - \frac{1}{4})$

solving $(i)$ and $(i i)$ we get $C (- \frac{3}{10}, - \frac{3}{5})$

Distance $B C = a$

$a = \sqrt{{(- \frac{3}{10} + 1)}^{2} + {(- \frac{3}{5} + \frac{1}{4})}^{2}} = \frac{7 \sqrt{5}}{20}$

Distance $C A = b$

$b = \sqrt{{(- \frac{3}{10} + 1)}^{2} + {(- \frac{3}{5} - \frac{1}{3})}^{2}} = \frac{7}{6}$

Distance $A B = c$

$c = \sqrt{{(- 1 + 1)}^{2} + {(\frac{1}{3} + \frac{1}{4})}^{2}} = \frac{7}{12}$

Coordiantes of incentre are
$(\frac{a x_{1} + b x_{2} + c x_{3}}{a + b + c}, \frac{a y_{1} + b y_{2} + c y_{3}}{a + b + c})$

$⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{(\frac{7 \sqrt{5}}{20} \times - 1) + (\frac{7}{6} \times - 1) + (\frac{7}{12} \times - \frac{3}{10})}{\frac{7 \sqrt{2}}{20} + \frac{7}{6} + \frac{7}{12}} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \frac{(\frac{7 \sqrt{5}}{20} \times \frac{1}{3}) + (\frac{7}{6} \times - \frac{1}{4}) + (\frac{7}{12} \times - \frac{3}{5})}{\frac{7 \sqrt{2}}{20} + \frac{7}{6} + \frac{7}{12}} ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦$

$(\frac{- 85 - 7 \sqrt{5}}{120}, \frac{21 \sqrt{5} - 65}{120})$

Radius is equal to distance from $x + 1 = 0$
$= ∣ ∣ ∣ ∣ ∣ ∣ \frac{\frac{- 85 - 7 \sqrt{5}}{120} + 1}{\sqrt{1^{2} + 0^{2}}} ∣ ∣ ∣ ∣ ∣ ∣$
$= \frac{35 - 7 \sqrt{5}}{120}$

Suggest Corrections

Similar questions

Q. Find the centre and radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are

3 x + 4 y + 2 = 0

3 x - 4 y + 12 = 0

, and

4 x - 3 y = 0

Q. Determine the nature of the quadrilateral formed by four lines

3 x + 4 y - 5 = 0; 4 x - 3 y - 5 = 0; 3 x + 4 y = 5 = 0

and

4 x - 3 y + 5 = 0

. Find the equation of the circle inscribed and circumscribing this quadrilateral.

Q. The equation of the circle inscribed in the triangle formed by the straight line

4 x + 3 y = 6

and both the coordinate axes is

Find the equation of circle circumscribing the quadrilateral formed by four lines $3 x + 4 y - 5 = 0$ , $4 x - 3 y - 5 = 0$ , $3 x + 4 y + 5 = 0$ and $4 x - 3 y + 5 = 0$

Q. Find the area of the triangle formed by the straight lines whose equations are

2 y + x - 5 = 0, y + 2 x - 7 = 0

, and

x - y + 1 = 0

Join BYJU'S Learning Program

Find the radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are 2x+4y+3=0, 4x+3y+3=0, and x+1=0.

Find the radius of the circle which is inscribed in the triangle formed by the straight lines whose equations are $2 x + 4 y + 3 = 0$ , $4 x + 3 y + 3 = 0$ , and $x + 1 = 0$ .