JEE MAIN - Mathematics (2021 - 27th July Evening Shift - No. 10)
Two sides of a parallelogram are along the lines 4x + 5y = 0 and 7x + 2y = 0. If the equation of one of the diagonals of the parallelogram is 11x + 7y = 9, then other diagonal passes through the point :
(1, 2)
(2, 2)
(2, 1)
(1, 3)
Explanation
Both the lines pass through origin.
_27th_July_Evening_Shift_en_10_1.png)
point D is equal to intersection of 4x + 5y = 0 & 11x + 7y = 9
So, coordinates of point $$D = \left( {{5 \over 3}, - {4 \over 3}} \right)$$
Also, point B is point of intersection of 7x + 2y = 0 and 11x + 7y = 9
So, coordinates of point $$B = \left( { - {2 \over 3},{7 \over 3}} \right)$$
diagonals of parallelogram intersect at middle let middle point of B, D
$$ \Rightarrow \left( {{{{5 \over 3} - {2 \over 3}} \over 2},{{{{ - 4} \over 3} + {7 \over 3}} \over 2}} \right) = \left( {{1 \over 2},{1 \over 2}} \right)$$
equation of diagonal AC
$$ \Rightarrow (y - 0) = {{{1 \over \alpha } - 0} \over {{1 \over \alpha } - 0}}(x - 0)$$
$$y = x$$
diagonal AC passes through (2, 2)
point D is equal to intersection of 4x + 5y = 0 & 11x + 7y = 9
So, coordinates of point $$D = \left( {{5 \over 3}, - {4 \over 3}} \right)$$
Also, point B is point of intersection of 7x + 2y = 0 and 11x + 7y = 9
So, coordinates of point $$B = \left( { - {2 \over 3},{7 \over 3}} \right)$$
diagonals of parallelogram intersect at middle let middle point of B, D
$$ \Rightarrow \left( {{{{5 \over 3} - {2 \over 3}} \over 2},{{{{ - 4} \over 3} + {7 \over 3}} \over 2}} \right) = \left( {{1 \over 2},{1 \over 2}} \right)$$
equation of diagonal AC
$$ \Rightarrow (y - 0) = {{{1 \over \alpha } - 0} \over {{1 \over \alpha } - 0}}(x - 0)$$
$$y = x$$
diagonal AC passes through (2, 2)
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