AB is a line of fixed length, 6 units, joining the points A(t,0) and B which lies on thepositive y axis. P is a point on AB distant 2 units from A. Express the coordinates of B and P in terms of t. Find the locus of P as t varies.

Asked by kandappan | 13th Feb, 2020, 01:39: PM

Expert Answer:

AB is a line segment of length 6 units where A is (t, 0) and B lies on the positive y-axis
Let B be (0, k) and the coordinates of P be (x, y)
As it is given that P is at a distance of 2 from A and lies on AB
Therefore, AP = 2 and PB = 4
So, we can say P divides the line segment AB in the ratio 1 : 2
rightwards double arrow x equals fraction numerator 1 left parenthesis 0 right parenthesis plus 2 left parenthesis t right parenthesis over denominator 1 plus 2 end fraction space space a n d space space y equals fraction numerator 1 left parenthesis k right parenthesis plus 2 left parenthesis 0 right parenthesis over denominator 1 plus 2 end fraction
rightwards double arrow x equals fraction numerator 2 t over denominator 3 end fraction space space a n d space space y equals k over 3
rightwards double arrow t equals fraction numerator 3 x over denominator 2 end fraction space space a n d space space k equals 3 y
A s space w e space k n o w space t h a t space A B equals 6
rightwards double arrow open parentheses t minus 0 close parentheses squared plus open parentheses 0 minus k close parentheses squared equals 36
rightwards double arrow t squared plus k squared equals 36
rightwards double arrow fraction numerator 9 x squared over denominator 4 end fraction plus 9 y squared equals 36
rightwards double arrow x squared plus 4 y squared equals 16

Answered by Renu Varma | 14th Feb, 2020, 11:52: AM

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