1. Tardigrade
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  4. Consider a variable line L which passes through the point of intersection ' P ' of the lines 3 x+4 y-12=0 and x+2 y-5=0, meeting the coordinate axes at the points A and B. Locus of the feet of the perpendicular from the origin on the variable line ' L ' has the equation

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Q. Consider a variable line $L$ which passes through the point of intersection ' $P$ ' of the lines $3 x+4 y-12=0$ and $x+2 y-5=0$, meeting the coordinate axes at the points $A$ and $B$.
Locus of the feet of the perpendicular from the origin on the variable line ' $L$ ' has the equation

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Solution:

Point of intersection of the line
$3x + 4y - 12 = 0$
$x + 2y - 5 = 0$
is $x = 2$ and $y = 3/2$
image
$\frac{k}{h} \cdot \frac{k-(3 / 2)}{h-2}=-1 $
$\frac{k}{h} \cdot \frac{2 k-3}{h-2}=-2$
$2 h(h-2)+k(2 k-3)=0 $
$2\left(x^2+y^2\right)-4 x-3 y=0$