Question

If a circle passes through the point \[( - 1,0)\] touches y- axis at \[(0,2)\], then the chord of the length of the circle along the x-axis is
A. \[\dfrac{5}{2}\]
B. \[5\]
C. \[\dfrac{3}{2}\]
D. \[3\]

Answer 1

Hint: First use the given data and draw the diagram of the circle and allocate all the given points and then form the equation of circle using given conditions and let it passes through the given point and then calculate the length of chord using distance formula among given points as \[d = \sqrt {{{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{x_2} - {x_1}} \right)}^2}} \]. The general equation of circle can be given as
\[{\left( {y - {y_1}} \right)^2} + {\left( {x - {x_1}} \right)^2} = {r^2}\]where \[\left( {{x_1},{y_1}} \right)\] are centres and r is the radius of circle.

Complete step by step Answer:

As the given information are as a circle passes through the point \[( - 1,0)\] touches y- axis at \[(0,2)\]
Diagram:

Hence, let the centre of the circle be \[\left( { - r,2} \right)\] and radius of the above circle be \[r\]. As the centre and the radius is known calculate the equation of circle as
\[{\left( {x + r} \right)^2} + {\left( {y - 2} \right)^2} = {r^2}\]
As point A \[( - 1,0)\] lies on the circle and so let satisfy the point in the equation of circle as
\[ \Rightarrow \]\[{\left( { - 1 + r} \right)^2} + {\left( {0 - 2} \right)^2} = {r^2}\]
On expanding the bracket and simplifying the terms
\[ \Rightarrow \]\[{r^2} - 2r + 1 + 4 = {r^2}\]
Hence, the value of r can be given as
\[
   \Rightarrow 2r = 5 \\
   \Rightarrow r = \dfrac{5}{2} \\
 \]
So, the centre of circle is given as \[C\left( { - \dfrac{5}{2},2} \right)\]
The general coordinate of the point M on the x axis can be given as \[M\left( { - \dfrac{5}{2},0} \right)\] and it is quite clear through the diagram.
As the length of chord is \[AB\]
It can be given as \[AB = AM + BM\]
And \[AM\] can be calculated using the distance formula \[d = \sqrt {{{\left( {{y_2} - {y_1}} \right)}^2} + {{\left( {{x_2} - {x_1}} \right)}^2}} \] as both points coordinates are known,
\[ \Rightarrow \]\[AM = \sqrt {{{\left( {0 - 0} \right)}^2} + {{\left( { - 1 - \left( { - \dfrac{5}{2}} \right)} \right)}^2}} \]
On simplifying further we cans see that,
\[ \Rightarrow \]\[AM = \sqrt {{{\left( {\dfrac{3}{2}} \right)}^2}} \]
Hence, on squaring and taking square root we can say that,
\[ \Rightarrow \]\[AM = \dfrac{3}{2}\]
As \[AM = BM\]
\[ \Rightarrow \]\[AB = 2AM\]
Hence, \[AB = 2 \times \dfrac{3}{2} = 3\]
Therefore, the length of chord is \[3\].
Hence, option (D) is our correct answer.

Note: The chord is a line segment that joins 2 points on the circumference of the circle. A chord only covers the part inside the circle. A chord of a circle is a straight line segment whose endpoints both lie on the circle. The infinite line extension of a chord is a secant line or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse.