The direction cosines of the normal drawn to the plane passing through the points \((2,-1,5),(1,-3,4),(5,2…
The direction cosines of the normal drawn to the plane passing through the points \((2,-1,5),(1,-3,4),(5,2,1)\) are
- \(\frac{11}{\sqrt{179}}, \frac{-7}{\sqrt{179}}, \frac{3}{\sqrt{179}}\)
- \(\frac{9}{\sqrt{134}}, \frac{-7}{\sqrt{134}}, \frac{2}{\sqrt{134}}\)
- \(\frac{11}{\sqrt{179}}, \frac{7}{\sqrt{179}}, \frac{-3}{\sqrt{179}}\)
- \(\frac{9}{\sqrt{134}}, \frac{7}{\sqrt{134}}, \frac{-2}{\sqrt{134}}\)
Solution
Let \(A=(2,-1,5)\)
and \(C=(5,2,1)\)
\(\therefore\) Equation of plane passing through \(A, B, C\) is
\(\begin{aligned}
&\left|\begin{array}{ccc}
x-x_1 & y-y_1 & z-z_1 \\
x_2-x_1 & y_2-y_1 & z_2-z_1 \\
x_3-x_1 & y_3-y_1 & z_3-z_1
\end{array}\right|=0 \\
& \Rightarrow \quad\left|\begin{array}{ccc}
x-2 & y+1 & z-5 \\
-1 & -2 & -1 \\
3 & 3 & -4
\end{array}\right|=0 \\
& 11 x-7 y+3 z-26=0
\end{aligned}\)
Required dc's are
\(\begin{aligned}
& =\frac{11}{\sqrt{121+49+9}}, \frac{-7}{\sqrt{121+49+9}}, \frac{3}{\sqrt{121+49+9}} \\
& =\left(\frac{11}{\sqrt{179}}, \frac{-7}{\sqrt{179}}, \frac{3}{\sqrt{179}}\right)
\end{aligned}\)
\(\therefore\) Hence, answer is (a).
Asked in: AP EAMCET 2019 (23 Apr Shift 1)
Practice more Three Dimensional Geometry questions on Aicharya