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Question: Find the equation of the circle and the length of its chord intercepted on the y-axis when a circle ...

Find the equation of the circle and the length of its chord intercepted on the y-axis when a circle contacts the x-axis at (3,0) and its radius is twice the radius of the circle x2+y2−2x−2y−2=0{x^2} + {y^2} - 2x - 2y - 2 = 0 ?

Explanation

Solution

We will first find the radius of the given circle using different formulas. We then by using the relation find the radius of the other circle. we will also find the center of the other circle to find its equation.

Complete step by step solution:
We have given the equation of the circle x2+y2−2x−2y−2=0{x^2} + {y^2} - 2x - 2y - 2 = 0
We will find the radius of the circle having equation x2+y2−2x−2y−2=0{x^2} + {y^2} - 2x - 2y - 2 = 0
We know that the general form of equation of circle when we have centre and its radius is (x−a)2+(y−b)2=r2{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2} where (a, b) is the centre and radius is r
We will rearrange the x2+y2−2x−2y−2=0{x^2} + {y^2} - 2x - 2y - 2 = 0 in the form (x−a)2+(y−b)2=r2{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} = {r^2}
The equation becomes
We have added 2 on both side of the equation
⇒x2−2x+1+y2−2y+1−2=2\Rightarrow {x^2} - 2x + 1 + {y^2} - 2y + 1 - 2 = 2
We know that (a−b)2=a2−2ab+b2{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}, we will us the formula and simplify the equation
⇒(x−1)2+(y−1)2=22\Rightarrow {\left( {x - 1} \right)^2} + {\left( {y - 1} \right)^2} = {2^2}
So, the radius is 2
So, the radius of our required circle is 4
We have given that the circle touches the x axis at (3, 0) so, the centre of the circle is (3, 4)
So, the equation of the circle having centre at (3, 4) and the radius is 4
⇒(x−3)2+(y−4)2=42\Rightarrow {\left( {x - 3} \right)^2} + {\left( {y - 4} \right)^2} = {4^2} (1)
We will now find the length of the cord intercept by Y axis
We pot x=0 in the equation 1
⇒9+(y−4)2=16\Rightarrow 9 + {\left( {y - 4} \right)^2} = 16
⇒(y−4)2=7\Rightarrow {\left( {y - 4} \right)^2} = 7
⇒y−4=±2.64\Rightarrow y - 4 = \pm 2.64
⇒y=6.64\Rightarrow y = 6.64 and y=1.35y = 1.35
So, the length of chord intercepted by Y axis is
⇒(6.64−1.35)=5.29\Rightarrow (6.64 - 1.35) = 5.29
Hence, the equation of circle is (x−1)2+(y−1)2=22{\left( {x - 1} \right)^2} + {\left( {y - 1} \right)^2} = {2^2} and length of chord is 5.29.

Note:
We should be familiar with the different forms of equations of a circle and how to find the equation of a circle when we have a center and its radius is given. To find the length of the chord by x axis we will put the value of x equals to zero and then find the distance between the values of y.