Question

A line 3 x+4y +10 = 0 cuts a chord of length 6 units on a circle with centre of the circle ( 2,1) . Find the equation of the circle in general form.​

Answer 1

Answer:

$x^2+y^2-4x-2y-20=0$

Step-by-step explanation:

A line 3 x+4y +10 = 0 cuts a chord of length 6 units on a circle with centre of the circle ( 2,1) . Find the equation of the circle in general form.​

Given: C( 2,1)

Let the chord be AB

Draw CM ⊥AB

Then AM=BM= 3 units

Now,

CM= Length of the perpendicuar from (2,1) to the line 3x+4y+10=0

$CM=|\frac{3(2)+4(1)+10}{\sqrt{3^2+4^2}}|$

$CM=|\frac{20}{\sqrt{25}}|$

$CM=|\frac{20}{5}|$

$CM=4 units$

In right angled triangle CMA

$CA^2=CM^2+AM^2$

$CA^2=4^2+3^2$

$CA^2=25$

$CA=5$

$\therefore\text{Radius of the circle}=5$

The equation of the circle is

$(x-h)^2+(y-k)^2=r^2$

$(x-2)^2+(y-1)^2=5^2$

$x^2+4-4x+y^2-2y+1-25=0$

$\implies\:\boxed{x^2+y^2-4x-2y-20=0}$