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  4. Let orthocentre of Δ ABC is (4,6) . If A=(4,7) and B=(- 2,4) , then coordinates of vertex C is

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Q. Let orthocentre of $\Delta ABC$ is $\left(4,6\right)$ . If $A=\left(4,7\right)$ and $B=\left(- 2,4\right)$ , then coordinates of vertex $C$ is

NTA AbhyasNTA Abhyas 2020Straight Lines

Solution:

Solution
Let feet of altitudes through $A,B,C$ are $D,E,F$ respectively.
Now, $C$ lies on the line $CF,CA$ & $CB$
Now, the slope of $AD$ is $\frac{7 - 6}{4 - 4}=N.D.$
$\Rightarrow $ the slope of $BC$ is $0$
$\Rightarrow $ equation of $BC$ is $y=4$
Now, the slope of $BE$ is $\frac{6 - 4}{4 - \left(\right. - 2 \left.\right)}=\frac{1}{3}$
$\Rightarrow $ the slope of $AC$ is $-3$
$\Rightarrow $ equation of the line $AC$ is $\left(y - 7\right)=-3\left(x - 4\right)$
Point of intersection of $y=4$ and $y-7=-3\left(x - 4\right)$ gives point $C$
$\Rightarrow C=\left(5 , 4\right)$