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9.Straight Line
hard
A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x-y+2=0$ and $y+2=0$, respectively. If the locus of the point $P$, that divides the $\operatorname{rod} AB$ internally in the ratio $2: 1$ is $9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0$, then $\alpha-\beta-\gamma$ is equal to :
A$24$
B$23$
C$21$
D$22$
(JEE MAIN-2025)
Solution
image$ h =\frac{3 \beta+\alpha}{3}$
$k =\frac{-4+\alpha+2}{3}$
$\alpha=3 k +2$
$2 \beta=3 h- a =3 h-3 k -2$
$\text { so } AB =8$
$(\alpha-\beta)^2+(\alpha+4)^2=64$
$\left(3 k +2-\left(\frac{3 h-3 k -2}{2}\right)\right)^2+(3 k +2+4)^2=64$
$\frac{(9 k -3 h+6)^2}{4}+(3 k +6)^2=64$
$9\left[(3 k – h +2)^2+4( k +2)^2\right]=64 \times 4$
$9\left( x ^2+13 y ^2-6 xy -4 x +28 y \right)=76$
$\alpha-\beta-\gamma=13+6+4=23$
Standard 11
Mathematics
