Straight Lines

Given the equation:

We can rewrite it as:

Now, we square both sides:

Expanding the right-hand side:

To find the point of interest, we need to find the intersection of the curve with the slopes of +1 and -1.

For the slope of +1, we have . Substituting this into the equation of the curve, we get:

Simplifying:

Rearranging, we find:

Squaring both sides, we get . Substituting this value back into the equation , we find .

So, for the slope of +1, the point of interest is .

For the slope of -1, we have . Substituting this into the equation of the curve, we get:

Simplifying:

Rearranging, we find:

However, the square root is defined only for non-negative values, so there is no solution for .

Therefore, there is no point of interest with a slope of -1.

Hence, the point on the curve at which the tangent is equally inclined to the axes is , which corresponds to option (B) .

1. Identify direction vectors:

For the first line:

For the second line:

2. Condition for perpendicularity:

Two lines are perpendicular if their direction vectors satisfy .

Compute the dot product:

3. Solve for :

Correct Answer: (A)