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Let A(z(1)), B(z(2)), C(z(3) and D(z(4))...

Let `A(z_(1)), B(z_(2)), C(z_(3)` and `D(z_(4))` be the vertices of a trepezium in an Argand plane such that `AB||CD` Let `|z_(1)-z_(2)|=4, |z_(3),z_(4)|=10` and the diagonals `AC` and `BD` intersects at `P`. It is given that `Arg((z_(4)-z_(2))/(z_(3)-z_(1)))=(pi)/2` and `Arg((z_(3)-z_(2))/(z_(4)-z_(1)))=(pi)/4`
Which of the following option(s) is/are incorrect?

A

Value of `|CP-DP|` is equal to `10/(sqrt(21))`

B

`PB:PD=2:5`

C

`PC:AC=5:7`

D

`PC:AC=2:3`

Text Solution

Verified by Experts

The correct Answer is:
D
`DeltaABP` and `DeltaCDP` are similar if `AP=2x`
`BP=2y` then `CP=5x, DP=5y`
Area of trapezium `ABCD=49/2xy`
`tan alpha=(2x)/(5y), tan beta =(2y)/(2x)` also `alpha+beta=45^(@)`
`impliesxy=10/21(x^(2)+y^(2))`
Also `AB^(2)=AP^(2)+BP^(2)impliesx^(2)+y^(2)=4impliesxy=40/21`
`Ar(DeltaPCD)=5xy`
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