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Which one of the following is the best condition for the plane \[ax+by+cz+d=0\] to intersect the x and y axes at equal angle?
(a) \[\left| a \right|=\left| b \right|\]
(b) a = -b
(c) a = b
(d) \[{{a}^{2}}+{{b}^{2}}=1\]

seo-qna
Last updated date: 11th Jun 2024
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Answer
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Hint: Find the point of intersection of the given plane on x and y – axis. To find this point of intersection on the x – axis, substitute the value of y and z coordinate equal to 0. Similarly, to find the point of intersection on y – axis, substitute the value of x and z coordinate equal to 0. To find the condition for equal angle, equate these x and y intercepts and find the relation between a and b.

Complete step-by-step answer:
Here, we have been provided with a plane \[ax+by+cz+d=0\] and we have been asked to find the condition such that this plane intersects x and y – axis at equal angle.
Now, let us find the point where this plane cuts the x and y – axis. We know that the point where the plane will cut the x – axis will have its value of y and x coordinate equal to 0. So, substituting the values y = 0 and z = 0 in the equation of plane, we get,
\[\Rightarrow ax+c=0\]
\[\Rightarrow x=\dfrac{-d}{a}\] - (1)
Similarly, the point where the plane will cut the y – axis will have its value of x and z coordinate equal to 0. So, substituting the values of x = 0 and z = 0 in the equation of plane, we get,
\[\Rightarrow by+d=0\]
\[\Rightarrow y=\dfrac{-d}{b}\] - (2)
Now, since the plane is intersecting the x and y axes at equal angle, that means x and y intercepts must be equal. So, from equation (1) and (2), we get,
\[\Rightarrow \dfrac{-d}{a}=\dfrac{-d}{b}\]
Cancelling the common factors and simplifying, we get,
\[\begin{align}
  & \Rightarrow \dfrac{1}{a}=\dfrac{1}{b} \\
 & \Rightarrow a=b \\
\end{align}\]
Since, we don’t know in which quadrant this intersection will take place, so a and b may be positive or negative depending on the quadrant. Therefore, we have to consider the modulus sign both the sides, we have,
\[\Rightarrow \left| a \right|=\left| b \right|\]
So, the correct answer is “Option (a)”.

Note: One may note that option (b) and option (c) are also correct, but they are actually a particular condition of option (a). The expression a = b denotes that the intersection will take place in either first or third quadrant while a = -b denotes that the intersection will take place in either second or fourth quadrant. But the expression \[\left| a \right|=\left| b \right|\] represents that intersection may take place in any of the four quadrants.