InterviewSolution
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InterviewSolution
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The line `y=(3x)/4`meets the lines `x-y=0`and `2x-y=0`at points `Aa n dB`, respectively. If `P`on the line `y=(3x)/4`satisfies the condition `P AdotP B=25 ,`then the number of possible coordinates of `P`is____ |
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Answer» Correct Answer - 3 Point P which lies on the line y=3x/4 can be chosen as P(h, 3h/4). If `theta` is the angle that the line y=3x/4 makes with the positive direction of the x-axis, then `"tan " theta = (3)/(4) " or cos " theta =(4)/(5) " and sin" theta = (3)/(5)` Now, the coordinates of points A and B which lie on the line y=3x/4 can be chosen as `A-=(h+(4r_(1))/(5), (3h)/(4) + (3r_(1))/(5)) " and "B-=(h+(4r_(2))/(5), (3h)/(4) + (3r_(2))/(5))` Since A lies on the line x-y+1 =0, we have `(h+(4r_(1))/(5)) -((3h)/(4) + (3r_(1))/(5)) +1 =0` `"or " r_(1) = (-5)/(4)(h+4)` Since B lies on the line 2x-y-5=0, we have `2(h+(4r_(2))/(5))- ((3h)/(4) + (3r_(2))/(5))-5=0` `"or " r_(2) = (-5)/(4)(h-4)` According to the given condition, we have `PA * PB =25` `i.e., |r_(1)| * |r_(2)| = 25` `i.e., (25)/(16)(h^(2) -16) = +- 25` `i.e., h^(2) = 16+-16 = 32, 0` `i.e., h=+-4sqrt(2), 0` Hence, the required points are (0, 0), `(4sqrt(2), 3sqrt(2)), " and " (-4sqrt(2), -3sqrt(2)).` |
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