Straight Lines Question - None

Question

Note that the lines $k_1 u - k_2 v = 0$ and $k_1 u + k_2 v = 0$ are equally inclined with the $uv$-axes. Hence the bisectors are $u = 0$ and $v = 0$. If the coordinate axes are directed along $u = 0$ and $v = 0$, which of the following are correct?

Accepted Answer

Step 1: Understanding the given lines

The lines k₁u - k₂v = 0 and k₁u + k₂v = 0 are equally inclined to the uv-axes, making equal angles with both coordinate axes.

Step 2: Identifying the bisectors

For lines of the form k₁u - k₂v = 0 and k₁u + k₂v = 0:

Adding: 2k₁u = 0 ⟹ u = 0

Subtracting: -2k₂v = 0 ⟹ v = 0

The angle bisectors are indeed u = 0 and v = 0.

Step 3: Coordinate transformation

When new coordinate axes are aligned with u = 0 and v = 0, we're essentially rotating the coordinate system by 45°. In the new coordinate system (let's call them x and y axes), the original pair of lines transforms.

Step 4: Form of transformed equations

Since the new x and y axes are the angle bisectors of the original pair, the new equation must represent a pair of lines symmetric about both new axes. The standard form is:

xy = constant or x² - y² = constant

Both represent rectangular hyperbolas with the coordinate axes as asymptotes—exactly what we expect when axes are angle bisectors of a pair of lines.

Options (a) and (d): These are likely xy = k and x² - y² = k forms respectively, which are the correct representations when the new axes become the bisectors of the original pair.

∴ Answer: (a) and (d)

Question:

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