Question

The length of the intercept on y-axis, by circle whose diameter is the line joining the points $$\left(-4,3\right)\mathrm{a}\mathrm{n}\mathrm{d}\left(12,-1\right)$$ is
A)$$3\sqrt{2}$$
B)$$\sqrt{13}$$
C)$$4\sqrt{13}$$
D)None of these.

Answer 1

Hint: First find the midpoint of 2 given points which will in turn become the center of the circle as 2 points are the endpoint of diameter. Find distance between 2 points center, any point to get the radius. As you know center and radius find the equation of circle. The y-intercept of the circle in the form of $$x^2+y^2+2gx+2fy+c=0\text{ is 2}\sqrt{f^2-c}$$.

Complete step by step solution:
The two given points of diameter are written as follows:
$$\mathrm{A}\left(-4,3\right);\mathrm{B}\left(12,-1\right)$$
Let the center of the circle be $$O=\left(x,y\right)$$point.
By above we can say the following statements:
x coordinate of the point denoted by A is given by -4.
x coordinate of the point denoted by B is given by 12.
x coordinate of the point denoted by O is given by x.
y coordinate of the point denoted by A is given by 3.
y coordinate of the point denoted by B is given by -1.
y coordinate of the point denoted by o is given by y.
The point O is the midpoint of points A, B.
The x coordinate of O is average of x coordinates of A, B, we get:
$$x=\text{ average of - 4, 12 = }{\displaystyle \frac{12-4}{2}}$$
By simplifying we get the value of x to be as:
$$x=4$$
The y coordinate of O is average of y coordinates of A, B, we get:
$$y=\text{ average of 3, - 1 = }{\displaystyle \frac{3-1}{2}}$$
By simplifying we get the value of y to be as:
$$y=1$$
So, the center of circle is given by point O $$\left(4,1\right)$$

The radius of the circle can be denoted as OA.
The distance between two points (a, b) (c, d) is d, can be given by:
$$d=\sqrt{{\left(a-c\right)}^2+{\left(b-d\right)}^2}$$
By substituting the values, we can write value of radius as:
Radius = distance between $$\left(4,1\right),\left(-4,3\right)=\sqrt{{\left(4+4\right)}^2+{\left(3-1\right)}^2}$$
By simplifying the above equation we can get value of radius as:
Radius $$=\sqrt{8^2+2^2}=\sqrt{64+4}=\sqrt{68}$$
Center $$=\left(4,1\right)$$
If center is (g, f) and radius r, we get equation as:
$${\left(x-g\right)}^2+{\left(y-f\right)}^2=r^2$$
By substituting the values, we get it as:
$${\left(x-4\right)}^2+{\left(y-1\right)}^2=68$$
By substituting $${\left(a-b\right)}^2=a^2+b^2-2ab,$$we get the equation as:
$$x^2+16-8x+y^2+1-2y=68$$
By simplifying the above equation, we get final equation as:
$$x^2+y^2-8x-2y-51=0$$
By comparing it to $$x^2+y^2+2gx+2fy+c=0$$, we get:
$$2g=-8,2f=-2\Rightarrow \mathrm{g}=-4,\mathrm{f}=-1,\mathrm{c}=-51$$
We know he y intercept given by:
$\text{y-intercept=2}\sqrt{f^2-c}$
By substituting f, c values, we get it as:
$\text{y-intercept=2}\sqrt{1-(-51)}=2\sqrt{52}$
52 can be written as $$13\times 4$$. So, by substituting it we get it as:
y intercept $$=4\sqrt{13.}$$
Therefore, option (c) is the correct answer.

Note: Be careful while getting the center as the whole equation of circle depends on that point. Don’t confuse between x, y coordinates. Alternate method is to substitute $$\mathrm{x}=0$$ and get the y values of the circle. Now get 2 intersection points on the y-axis. The distance between the two points is called the y-intercept. Anyway you get the same result.