Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7}$

Solution

Let the equation of the plane be ax + by + cz + d = 0 ........(i)

Since the plane passes through the point A ( 0, 0, 0 ) and B ( 3, -1, 2),

we have

a x 0 + b x 0 + c x 0 + d = 0

$\Rightarrow$ d = 0 ................(ii)

Similarly for point B ( 3, -1, 2 ), a x 3 + b x ( - 1 ) + c x 2 + d = 0

3a - b + 2c = 0 ( Using , d = 0 ) ............(iii)

$Given equation of the line is \frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7} We can also write the above equation as \frac{x - 4}{1} = \frac{y - (- 3)}{- 4} = \frac{z - (- 1)}{7}$

The required plane is parellel to the above line .

Therefore, a x 1 + b x ( - 4 ) + c x 7 = 0

$\Rightarrow$ a - 4b + 7c = 0 ............(iv)

Cross multiplying equations (iii) and (iv), we obtain:

$\frac{a}{(- 1) x 7 - (- 4) x 2} = \frac{b}{2 x 1 - 3 x 7} = \frac{c}{3 x (- 4) - 1 x (- 1)} \Rightarrow \frac{a}{- 7 + 8} = \frac{b}{2 - 21} = \frac{c}{- 12 + 1} \Rightarrow \frac{a}{1} = \frac{b}{- 19} = \frac{c}{- 11} = k \Rightarrow a = k, b = - 19 k, c = - 11 k$

Substituting the values of a, b and c in equation ( 1 ), we obtain the equation of plane as:

kx - 19ky - 11kz + d = 0

$\Rightarrow$ k ( x - 19y - 11z ) = 0 ..........( From equation (ii) )

$\Rightarrow$ x - 19y - 11z = 0

So, the equation of the required plane is x - 19y - 11z .

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Three Dimensional Geometry

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7}$

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

Mock Test Series

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Entrance Exams Preparation

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Three Dimensional Geometry

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line x - 41 = y + 3-4 = z + 17

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Find the Cartesian equation of the plane passing through the points A(0, 0, 0) and B(3, -1, 2) and parallel to the line $\frac{x - 4}{1} = \frac{y + 3}{- 4} = \frac{z + 1}{7}$