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Q.
The coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$ are
Straight Lines
Solution:
Let $(h, k)$ be the coordinates of the foot of the perpendicular from the point $(2,3)$ on the line $x+y-11=0$. Then, the slope of the perpendicular line is $\frac{k-3}{h-2}$. Again, the slope of the given line $x+y-11=0$ is $-1$.
Using the condition of perpendicularity of lines, we have
$\left(\frac{k-3}{h-2}\right)(-1)=-1 \left(\because m_1 m_2=-1\right)$
or $ k-h=1 ....$(i)
Since, $(h, k)$ lies on the given line, so we have
$h+k-11 =0 $
or $h+k =11 ....$(ii)
On solving Eqs. (i) and (ii), we get $h=5$ and $k=6$
Thus, $(5,6)$ are the required coordinates of the foot of perpendicular.