Advertisements
Advertisements
Question
Find the equation of the plane through the line of intersection of the planes \[x + y + z =\]1 and 2x \[+\] 3 \[+\] y \[+\] 4\[z =\] 5 and twice of its \[y\] -intercept is equal to three times its \[z\]-intercept
Solution
The equation of the family of the planes passing through the intersection of the planes x+ y + z = 1 and 2x + 3y + 4z = 5 is
(x + y + z − 1) + k(2x + 3y + 4z − 5) = 0, where k is some constant
⇒ (2k + 1)x + (3k + 1)y + (4k + 1)z = 5k + 1 .....(1)
\[ \Rightarrow \left( 5k + 1 \right)\left( 8k + 2 - 9k - 3 \right) = 0\]
\[ \Rightarrow \left( 5k + 1 \right)\left( - k - 1 \right) = 0\]
\[ \Rightarrow \left( 5k + 1 \right)\left( k + 1 \right) = 0\]
\[ \Rightarrow k = - \frac{1}{5} or k = - 1\]
\[ \Rightarrow 3x + 2y + z = 0\]
This plane passes through the origin. So, the intercepts made by the plane with the coordinate axes is 0. Hence, this equation of plane is not accepted as twice of y-intercept is not equal to three times its z-intercept.
Putting \[k = - 1\] in (1), we get
\[ \Rightarrow - x - 2y - 3z = - 4\]
\[ \Rightarrow x + 2y + 3z = 4\]
Thus, the equation of the required plane is x + 2y + 3z = 4.
APPEARS IN
RELATED QUESTIONS
Write the sum of intercepts cut off by the plane `vecr.(2hati+hatj-k)-5=0` on the three axes
Find the intercepts cut off by the plane 2x + y – z = 5.
Prove that if a plane has the intercepts a, b, c and is at a distance of P units from the origin, then `1/a^2 + 1/b^2 + 1/c^2 = 1/p^2`
Write the equation of the plane whose intercepts on the coordinate axes are 2, −3 and 4.
Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
2x + 3y − z = 6
Reduce the equations of the following planes to intercept form and find the intercepts on the coordinate axes.
2x − y + z = 5
Find the equation of a plane which meets the axes at A, B and C, given that the centroid of the triangle ABC is the point (α, β, γ).
Find the equation of the plane with intercept 3 on the y-axis and parallel to the ZOX plane.
Find the equation of the plane through the point \[2 \hat{i} + \hat{j} - \hat{k} \] and passing through the line of intersection of the planes \[\vec{r} \cdot \left( \hat{i} + 3 \hat{j} - \hat{k} \right) = 0 \text{ and } \vec{r} \cdot \left( \hat{j} + 2 \hat{k} \right) = 0 .\]
Find the equation of the plane passing through the line of intersection of the planes 2x − y = 0 and 3z − y = 0 and perpendicular to the plane 4x + 5y − 3z = 8
Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z − 4 = 0 and 2x + y − z + 5 = 0 and which is perpendicular to the plane 5x + 3y − 6z+ 8 = 0.
Find the equation of the plane through the line of intersection of the planes x + 2y + 3z + 4 = 0 and x − y + z + 3 = 0 and passing through the origin.
Find the equation of the plane that is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z − 4 = 0, 2x + y − z + 5 = 0.
Find the equation of the plane passing through the intersection of the planes 2x + 3y − z+ 1 = 0 and x + y − 2z + 3 = 0 and perpendicular to the plane 3x − y − 2z − 4 = 0.
Find the equation of the plane through the line of intersection of the planes \[\vec{r} \cdot \left( \hat{i} + 3 \hat{j} \right) + 6 = 0 \text{ and } \vec{r} \cdot \left( 3 \hat{i} - \hat{j} - 4 \hat{k} \right) = 0,\] which is at a unit distance from the origin.
Find the equation of the plane that contains the line of intersection of the planes \[\vec{r} \cdot \left( \hat{i} + 2 \hat{j} + 3 \hat{k} \right) - 4 = 0 \text{ and } \vec{r} \cdot \left( 2 \hat{i} + \hat{j} - \hat{k} \right) + 5 = 0\] and which is perpendicular to the plane \[\vec{r} \cdot \left( 5 \hat{i} + 3 \hat{j} - 6 \hat{k} \right) + 8 = 0 .\]
Find the equation of the plane passing through the intersection of the planes \[\vec{r} \cdot \left( 2 \hat{i} + \hat{j} + 3 \hat{k} \right) = 7, \vec{r} \cdot \left( 2 \hat{i} + 5 \hat{j} + 3 \hat{k} \right) = 9\] and the point (2, 1, 3).
A plane makes intercepts −6, 3, 4 respectively on the coordinate axes. Find the length of the perpendicular from the origin on it.
Find the equation of the plane through the intersection of the planes 3x − y + 2z = 4 and x + y + z = 2 and the point (2, 2, 1).
Find the vector equation of the plane passing through the intersection of the planes
\[\vec{r} \cdot \left( \hat{ i } + \hat{ j }+ \hat{ k }\right) = \text{ 6 and }\vec{r} \cdot \left( \text{ 2 } \hat{ i} +\text{ 3 } \hat{ j } + \text{ 4 } \hat{ k } \right) = - 5\] and the point (1, 1, 1).
Find the equation of the plane which contains the line of intersection of the planes x \[+\] 2y \[+\] 3 \[z - \] 4 \[=\] 0 and 2 \[x + y - z\] \[+\] 5 \[=\] 0 and whose x-intercept is twice its z-intercept. Hence, write the equation of the plane passing through the point (2, 3, \[-\] 1) and parallel to the plane obtained above.
The intercepts made by the plane 2x – 3y + 5z + 4 = 0 on the coordinate axes are `-2, 4/3, (-4)/5`.
The intercepts made on the coordinate axes by the plane 2x + y − 2z = 3 are:
