Show that the equation ax+by+d=0 represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points A(2,−3,1) and B(−4,7,6).

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Filo · Accepted Answer

The given equation is ax+by+0.z+d=0 which is of the form ax+by+cz+d=0.Therefore, it represents a plane.Direction ratio of normal to the plane are a,b,0Direction ratio of the z-axis are 0,0,1Now, a×0+b×0+0×1=0This shows that the given plane is parallel to the z-axis.Let the required plane be ax+by+d=0 ..........(i)Since it passes through the points A(2,−3,1) and B(−4,7,6), we have2a−3b+d=0 ........(ii)−4a+7b+d=0 .........(iii)On solving (ii) and (iii) by cross multiplication, we get(−3−7)a​=(−4−2)b​=(14−12)c​⇒−10a​=−6b​=2c​⇒5a​=3b​=−1c​=k (say)∴a=5k,b=3k and c=−kPutting these values in (i), we get5kx+3ky−k=0 ⇒5x+3y−1=0,which is the required equation of the plane.

Question Text	Show that the equation $a x + b y + d = 0$ represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points $A (2, - 3, 1)$ and $B (- 4, 7, 6)$ .
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Show that the equation $a x + b y + d = 0$ represents a plane parallel to the z-axis. Hence, find the equation of a plane which is parallel to the z-axis and passes through the points $A (2, - 3, 1)$ and $B (- 4, 7, 6)$ .

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