Find the locus of a point such that the line segments having end points (2, 0) and (-2, 0) subtend a right angle at that point.
Key points to solve the problem:
• Idea of distance formula- Distance between two points A(x1,y1) and B(x2,y2) is given by- AB =
• Pythagoras theorem: In right triangle ΔABC : the sum of the square of two sides is equal to the square of its hypotenuse.
How to approach: To find the locus of a point we first assume the coordinate of the point to be (h, k) and write a mathematical equation as per the conditions mentioned in the question and finally replace (h, k) with (x, y) to get the locus of the point.
Let the coordinates of a point whose locus is to be determined to be (h, k) and name the moving point to be C.
According to a question on drawing the figure, we get a right triangle Δ ABC.
From Pythagoras theorem we have:
BC2 + AC2 = AB2
From distance formula:
BC =
AC =
And AB = 4
∴
⇒
⇒
⇒ 2h2 + 2k2 – 8 = 0
⇒ h2 + k2 = 4
Replace (h,k) with (x,y)
Thus, the locus of a point is x2 + y2 = 4 ….ans