Question:
If \((a,b,c)\) are the direction ratios of a line joining the points \((4,3,-5)\) and \((-2,1,-8)\), then the point \(P(a,3b,2c)\) lies on the plane
If \((a,b,c)\) are the direction ratios of a line joining the points \((4,3,-5)\) and \((-2,1,-8)\), then the point \(P(a,3b,2c)\) lies on the plane
Show Hint
Direction ratios of the line joining \((x_1,y_1,z_1)\) and \((x_2,y_2,z_2)\) are \((x_2-x_1,\;y_2-y_1,\;z_2-z_1)\).
Updated On: Jun 15, 2026
- \(x+y+z=0\)
- \(x+y-2z=0\)
- \(x+2y+3z=0\)
- \(x-2y+3z=0\)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Step 1: Find the direction ratios of the line.
The given points are
\[ A(4,3,-5) \] and
\[ B(-2,1,-8) \]
Direction ratios are obtained by subtracting corresponding coordinates:
\[ a=-2-4=-6 \] \[ b=1-3=-2 \] \[ c=-8-(-5)=-3 \]
Thus, one set of direction ratios is
\[ (a,b,c)=(-6,-2,-3) \]
Step 2: Find the coordinates of point \(P\).
Given,
\[ P(a,3b,2c) \]
Substitute the values of \(a,b,c\):
\[ P=(-6,3(-2),2(-3)) \]
\[ P=(-6,-6,-6) \]
Step 3: Check which plane satisfies the point.
For plane
\[ x+y-2z=0, \] substitute \((-6,-6,-6)\):
\[ -6+(-6)-2(-6) \]
\[ =-12+12 \]
\[ =0 \]
Hence, the point satisfies the plane equation.
Step 4: Final conclusion.
Therefore, the point \(P(a,3b,2c)\) lies on the plane
\[ \boxed{x+y-2z=0} \]
The given points are
\[ A(4,3,-5) \] and
\[ B(-2,1,-8) \]
Direction ratios are obtained by subtracting corresponding coordinates:
\[ a=-2-4=-6 \] \[ b=1-3=-2 \] \[ c=-8-(-5)=-3 \]
Thus, one set of direction ratios is
\[ (a,b,c)=(-6,-2,-3) \]
Step 2: Find the coordinates of point \(P\).
Given,
\[ P(a,3b,2c) \]
Substitute the values of \(a,b,c\):
\[ P=(-6,3(-2),2(-3)) \]
\[ P=(-6,-6,-6) \]
Step 3: Check which plane satisfies the point.
For plane
\[ x+y-2z=0, \] substitute \((-6,-6,-6)\):
\[ -6+(-6)-2(-6) \]
\[ =-12+12 \]
\[ =0 \]
Hence, the point satisfies the plane equation.
Step 4: Final conclusion.
Therefore, the point \(P(a,3b,2c)\) lies on the plane
\[ \boxed{x+y-2z=0} \]
Was this answer helpful?
0
0
Top AP EAMCET Mathematics Questions
- If $\int \cos x . \cos 2x . \cos 5x dx = A \; \sin 2x + B \sin 4x + C \sin 6x + D \sin 8x + k $ (where $k$ is the arbitrary constant of integration), then $\frac{1}{B} + \frac{1}{C} = $
- If $k$ is one of the roots of the equation $x^2 - 25x + 24 = 0 $ such that $A = \begin{bmatrix}1&2&1\\ 3&2&3\\ 1&1&k\end{bmatrix} $ is a non-singular matrix, then $A^{-1}$ =
- If $p$ and $q$ are respectively the global maximum and global minimum of the function $f(x) = x^2 e^{2x}$ on the interval $[-2, 2]$ , then $pe^{-4} + qe^4 = $
- If $\displaystyle\sum^n_{k - 1} \tan^{-1} \left( \frac{1}{k^2 + k + 1} \right) = \tan^{-1} (\theta) $ , then $\theta$ =
- If the line joining the points $A(\alpha)$ and $B(\beta)$ on the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$ is a focal chord, then one possible values of $\cot \frac{\alpha}{2} . \cot \frac{\beta}{2}$ is
View More Questions
Top AP EAMCET 3D Geometry Questions
- The length of the internal bisector of angle A in \( \triangle ABC \) with vertices \( A(4,7,8) \), \( B(2,3,4) \), and \( C(2,5,7) \) is:
- If a line \( L \) makes angles \( \frac{\pi}{3} \) and \( \frac{\pi}{4} \) with the Y-axis and Z-axis respectively, then the angle between \( L \) and another line having direction ratios \( 1,1,1 \) is:
- If \( l, m, n \) are the direction cosines of a line that is perpendicular to the lines having the direction ratios \( (1,2,-1) \) and \( (-2,1,1) \), then \( (l+m+n)^2 \) =
- The foot of the perpendicular drawn from a point \( A(1,1,1) \) onto a plane \( \pi \) is \( P(-3,3,5) \). If the equation of the plane parallel to the plane \( \pi \) and passing through the midpoint of \( AP \) is \[ ax - y + cz + d = 0, \] then \( a + c - d \) is:
- The distance of a point \( (2,3,-5) \) from the plane \( \vec{r} \cdot (4i - 3j + 2k) = 4 \) is:
Top AP EAMCET Questions
- If $\int \cos x . \cos 2x . \cos 5x dx = A \; \sin 2x + B \sin 4x + C \sin 6x + D \sin 8x + k $ (where $k$ is the arbitrary constant of integration), then $\frac{1}{B} + \frac{1}{C} = $
- If $k$ is one of the roots of the equation $x^2 - 25x + 24 = 0 $ such that $A = \begin{bmatrix}1&2&1\\ 3&2&3\\ 1&1&k\end{bmatrix} $ is a non-singular matrix, then $A^{-1}$ =
- If only $\frac{1}{51}^{th}$ ot the main current is to be passed through a galvanometer then the shunt required is $R_1$ and if only $\frac{1}{11}^{th}$ of the main voltage is to be developed across the galvanometer, then the resistance required is $R_2$. Then $\frac{R_2}{R_1} =$
- If $p$ and $q$ are respectively the global maximum and global minimum of the function $f(x) = x^2 e^{2x}$ on the interval $[-2, 2]$ , then $pe^{-4} + qe^4 = $
- If $\displaystyle\sum^n_{k - 1} \tan^{-1} \left( \frac{1}{k^2 + k + 1} \right) = \tan^{-1} (\theta) $ , then $\theta$ =
View More Questions