Question

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

Answer 1

Hint: Find midpoints of diameter to find center of a circle. Then find diameter with the help of distance formula and at least use y intercept formula to find y coordinates.
The lengths of intercepts made by the circle \[{x^2} + {y^2} + 2gx + 2fy + c = 0\] with x and y axes are $2\sqrt {{g^2} - c} $ and $2\sqrt {{f^2} - c} $ respectively.

Complete step-by-step answer:
The two given points of a diameter are (−4,3) and (12,−1).
If the endpoints of one diameter of a circle are \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( {{x_2},{\text{ }}{y_2}} \right)\]
Then, the center of the circle has the coordinates:-
$\left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) \Rightarrow \left( {\dfrac{{ - 4 + 12}}{2},\dfrac{{3 - 1}}{2}} \right) \Rightarrow \left( {4,1} \right)$……………(1)

So, the center of the circle is given by point (4, 1)
Use the Distance Formula to find the diameter and radius of the circle.
\[
  \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} = d \\
   \Rightarrow \sqrt {{{(12 - ( - 4))}^2} + {{(( - 1) - 3)}^2}} = d \\
   \Rightarrow \sqrt {{{16}^2} + {4^2}} = d \\
   \Rightarrow d = \sqrt {272} \\
   \Rightarrow r = \dfrac{{\sqrt {272} }}{2} = \sqrt {68} \\
 \]
Diameter, d = \[\sqrt {272} \] whereas radius, r = \[\sqrt {68} \]………………….(2)
A method is to attempt to reconstruct the equation of the circle in the form
\[{(x - a)^2} + {(y - b)^2} = {r^2}\] by completing the square.
Using center of the circle and radius from the equation (1) and (2):-
\[{(x - 4)^2} + {(y - 1)^2} = 68\]
We start by collecting together the terms in x, and the terms in y.
So we rewrite our equation after opening the brackets and simplifying the equation
\[{x^2} + {y^2} - 8x - 2y - 51 = 0\]
Here g= -4, f = -1 and c = -51
Y intercept is equal to = $2\sqrt {{f^2} - c} = 2\sqrt {{{( - 1)}^2} - ( - 51)} = 2\sqrt {52} = 4\sqrt {13} $

So, 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 while substituting x and y values.
Usually in these types of questions, you are given the equation of the circle C and the equation of the line L. But here we are not given both of them but given the coordinates of a diameter which can provide the center and the radius of the circle. So, ultimately provided the equation of the circle.