Question

Let \(\mathbf{a} = \mathbf{i} + 2\mathbf{j} - \mathbf{k}\) and \(\mathbf{b} = 6\mathbf{i} - \mathbf{j} + 2\mathbf{k}\) be two vectors. If \[ |\mathbf{a} \times \mathbf{b}|^2 + |\mathbf{a} . \mathbf{b}|^2 = f(x,y)(x+y) - 46 = 0, \] then what does this represent?

• A. A pair of lines
• B. An ellipse
• C. A hyperbola
• D. A circle

Hint:

Use vector identities and recognize quadratic forms to identify conic sections from given expressions.

Answer 1

The Correct Option is D

Recall the vector identities: \[ |\mathbf{a} \times \mathbf{b}|^2 + (\mathbf{a} . \mathbf{b})^2 = |\mathbf{a}|^2 |\mathbf{b}|^2. \] Given that the expression equals zero when \(f(x,y)(x+y) - 46 = 0\), this represents a quadratic form of \(x\) and \(y\). The standard form of a circle is \(Ax^2 + Ay^2 + \ldots = \text{constant}\), which fits the given form. Hence, the equation represents a circle.