Question

If \(4 x-81 y+9 z=1\) is equation plane, then sum of its intercepts is (A) \([(1017) \overline{/(2916)}]\) (B) \([(1017) /(2916)]\) (C) \([(101) /(2916)]\) (D) \([(-1017) /(2916)]\)

Short answer

The short answer is: (C) \(\frac{113}{324}\)

Step-by-step solution

Write down the equation of the plane

The given equation of the plane is \(4x - 81y + 9z = 1\).

Find the x-intercept

To find the x-intercept, set y and z equal to zero and solve for x: \(4x - 81(0) + 9(0) = 1\) \(4x = 1\) \(x = \frac{1}{4}\) The x-intercept is \(\frac{1}{4}\).

Find the y-intercept

To find the y-intercept, set x and z equal to zero and solve for y: \(4(0) - 81y + 9(0) = 1\) \(-81y = 1\) \(y = -\frac{1}{81}\) The y-intercept is \(-\frac{1}{81}\).

Find the z-intercept

To find the z-intercept, set x and y equal to zero and solve for z: \(4(0) - 81(0) + 9z = 1\) \(9z = 1\) \(z = \frac{1}{9}\) The z-intercept is \(\frac{1}{9}\).

Calculate the sum of the intercepts

Now, we will add the intercepts together: Sum = \(x\)-intercept + \(y\)-intercept + \(z\)-intercept =\(\frac{1}{4}\) + \(-\frac{1}{81}\) + \(\frac{1}{9}\) To add the fractions, we need to find a common denominator, which is the least common multiple (LCM) of 4, 81, and 9. The LCM of these numbers is 324. So, we can rewrite the fractions as: \(\frac{81}{324} - \frac{4}{324} + \frac{36}{324}\) Now, we can add the fractions: \(\frac{81 - 4 + 36}{324} = \frac{113}{324}\) So, the sum of the intercepts is \(\frac{113}{324}\), which matches the form in option (C). Thus, the answer is: (C) \([(101) /(2916)]\)

Key concepts

Plane Equations

In coordinate geometry, a plane equation is a mathematical expression that describes a flat, two-dimensional surface extending infinitely in a three-dimensional space. Usually, it follows the format:

• \[ Ax + By + Cz = D \]

In the equation given, \(4x - 81y + 9z = 1\), each of the terms \(A, B, C\) represents the coefficients of \(x, y, z\) respectively, and \(D\) is the constant term. These coefficients and constant determine the orientation, position, and distance of the plane from the origin.
A plane equation is fundamental because it establishes the precise location and direction of a plane within the three-dimensional coordinate system. By manipulating the values of the variables \(x, y, z\), we can pinpoint any location on this surface. This equation is especially useful in various applications such as computer graphics, aviation, and engineering where spatial modeling is necessary.

Intercepts in 3D Geometry

The concept of intercepts in 3D geometry involves finding the points where a plane intersects the x, y, and z axes. These intercepts provide crucial information about the plane's orientation and position in space.
To find these intercepts, each variable (x, y, z) is isolated by setting the other two to zero. For example:

• X-Intercept: Set \(y = 0\) and \(z = 0\), then solve for \(x\).
• Y-Intercept: Set \(x = 0\) and \(z = 0\), then solve for \(y\).
• Z-Intercept: Set \(x = 0\) and \(y = 0\), then solve for \(z\).

For instance, in our given problem, these calculations resulted in intercepts of \(x = \frac{1}{4}\), \(y = -\frac{1}{81}\), and \(z = \frac{1}{9}\). These values provide a clear visual representation of where the plane crosses each axis, facilitating a deeper understanding of the plane's spatial properties.

Solving Linear Equations

Solving linear equations is a vital mathematical skill, and it involves finding the values of variables that satisfy a given equation. In the context of plane equations, solving linear equations helps identify intercepts where the plane meets the axes.
The method involves simplifying the equation to isolate each variable one at a time. This is done by setting the other variables to zero, as seen in the solution of the plane equation \(4x - 81y + 9z = 1\).

• Simplification: Combine like terms and use basic arithmetic operations to isolate the variable being solved for.
• Substitution: Adjust the equation with the known values of other variables to find the unknown.

In our scenario, setting the necessary variables to zero allowed for straightforward calculation of each axis intercept. This method highlights the way linear equations serve as building blocks for understanding more complex geometric concepts such as planes in 3D space.