If a line segment AM=a moves in the plane XOY remaining parallel to OX so that the left end point A slides along the circle \[{x^2} + {y^2} = {a^2}\], the locus of M is
(A) \[{x^2} + {y^2} = 4{a^2}\]
(B) \[{x^2} + {y^2} = 2ax\]
(C) \[{x^2} + {y^2} = 2ay\]
(D) \[{x^2} + {y^2} - 2ax - 2ay = 0\]
VerifiedHint:
According to the given question, firstly draw a diagram corresponding to given values.
Then, calculate the coordinates of the locus M \[(p,q)\] corresponding to the A \[(x,y)\]. Hence, substitute the value in the given circle equation.
Complete step by step solution:
Firstly, we construct a diagram with the help of given points.
1) Construct a circle with centre O and with the help of the equation \[{x^2} + {y^2} = {a^2}\] as shown in figure.
2) Then, draw a parallel line that is AM \[||\] OX as shown in figure.
Let us assume the coordinates of A \[(x,y)\] and M \[(p,q)\] .
Since we are given that always AM is parallel to OX that is AM \[||\] OX .
Therefore, we can calculate the values of p and q .
As, it is given that the length of AM is a.
Hence, coordinates of p = \[x + AM = x + a\] and coordinates of q = y .
So, from the above equations we can calculate the value of x that is \[x = p - a\] and \[y = q\].
As we know A \[(x,y)\] lies on the circle \[{x^2} + {y^2} = {a^2}\] - equation 1
So, now we will substitute the values of x and y in equation 1.
Hence, we get \[{\left( {p - a} \right)^2} + {q^2} = {a^2}\]
Opening \[{\left( {p - a} \right)^2}\] with the help of algebraic identity that is \[{\left( {a - b} \right)^2} = {a^2} + {b^2} - 2ab\]. So,
\[{p^2} + {a^2} - 2ap + {q^2} = {a^2}\]
Cancelling \[{a^2}\] from both left hand side and right hand side we get,
\[{p^2} - 2ap + {q^2} = 0\]
On rearranging the equation:
\[{p^2} + {q^2} = 2ap\]
As, we have calculated the locus of M \[(p,q)\] that is \[{x^2} + {y^2} = 2ax\]
Hence, option (B) \[{x^2} + {y^2} = 2ax\] is correct.
Note:
To solve these types of questions, we must draw the diagram to make the question easier to understand. While finding the locus do not forget to put the values given point which lies on the circle.
