Straight Lines

To solve this problem, we need to determine the nature of the tangents to the given conic section equation at the points where it crosses the X-axis.

The curve is given by the equation . This equation represents a general conic section.

The condition for the curve to cross the X-axis is . Substituting into the equation gives us:

Thus, the points where the curve crosses the X-axis are and where and .

Now, we find the equation of the tangents at these points. The general formula for the tangent to a conic at a point is:

.

For the point : Substitute , the tangent equation becomes:

For the point : Substitute , the tangent equation becomes:

Both equations are of the form and respectively.

These equations show that both tangents are vertical lines, which means they are parallel to each other. Therefore, the two tangents at the points where the curve crosses the X-axis are parallel.

Hence, the correct answer is: Parallel.

The problem requires us to find the equations of straight lines that pass through the given point and have the sum of their intercepts on the coordinate axes equal to -1.

A line in the intercept form is given by the equation: , where and are the x-intercept and y-intercept, respectively.

According to the problem, these intercepts have a sum equal to -1:

\) .

Since the line passes through the point (4, 3), substituting these coordinates into the line's equation, we have:

.

Now, we can solve these equations:

1. Equation 1:
2. Equation 2:

We assume possible values for and that satisfy both equations. Solving these simultaneously, we try different scenarios:

Case 1:

1. If and :
   • Substituting in Equation 1: , so satisfies Equation 1.
   • Substituting in Equation 2: , satisfies Equation 2.
2. Thus, the line equation is: .

Case 2:

1. If and :
   • Substituting in Equation 1: , satisfies Equation 1.
   • Substituting in Equation 2: , satisfies Equation 2.
2. Thus, the line equation is: .

Therefore, the correct answer is Option 1: and .