The line $4 x + 3 y – 4 = 0 $ divides the circumference of the circle centred at $ (5,3)$ in the ratio $1:2$ . Then the equation of the circle is?
I tried to solve this problem my taking parametric points but was confused which distance should I take please help me out.
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We have $4x+3y=4$, so $y=\frac{-4x}{3}+\frac{4}{3}$.
The slopes of perpendicular lines produce $-1$ when multiplied. The line's slope is $\frac{-4}{3}$, so the slope of the perpendicular is $\frac{3}{4}$.
We know $(5,3)$ is on the line.
So $(y-3)=(x-5)\frac{3}{4}$ gives you the equation of the perpendicular.
So $$(x-5)\frac{3}{4}+3=\frac{-4}{3}x+\frac{4}{3}$$
$$\frac{25}{12}x=\frac{25}{12}\implies x=1; y=0 $$
The distance betwen the line and the center is $\sqrt{(5-1)^2+(3-0)^2}=5$
The chords are in a 2:1 ratio. Let $p$ be the radian measure of the shorter, then the other has radian measure $2p$. $3p=2\pi$, so $p=2\pi/3$.
The radius is the hypotenuse of a right triangle made from the line, radius, and the perpendicular. The length of perpendicular between line and center is cosine of half $p$ times the radius. We know the length is $5$ and the cosine is $\frac{1}{2}$, so the radius must be $5$.
$$(x-5)^2+(y-3)^2=10^2=x^2+y^2-10x -6y-66=0$$
Since, given that $4x+3y-4=0$ divides the circumference of the circle in the ratio of $1:2$, the subtended angle at the center is $\dfrac{2\pi}{3}$
The perpendicular distance from the circle of the given line is $5$.
The radius is $10$
So, the equation of the circle is $x^2+y^2-10x-6y-66=0$