Straight Lines And Pair Of Straight Lines

Let's analyze each statement to determine which one is not correct.Statement A"If passes through two points of , then must be rational." consists of points where both and are rational numbers. If passes through two such points ( ) and ( ), the slope of the line would be:$ a_1, a_2, b_1 b_2 m l S m m c l l m m c l S (a, b) a, b \in \mathbb{Q}, b=m a+c m c a, m a c m a b l m c l S y=m x+c (a, b) a, b \in \mathbb{Q} b=m a+c m m a a c m a+c b b \in \mathbb{Q} l S \mathbf{B} m c l S$.

To find the intersections of the given lines in the XY-plane, let's analyze each line and express them in the slope-intercept form .Lines and Their EquationsLine :Given by the equation .Rearranging, we get: Line :Passes through the points and .Slope is calculated as: Using point , the line equation is: Line :Passes through and is parallel to the vector .The slope of the vector, and hence the line, is: Using point , the line equation becomes: Solving for IntersectionsNext, we find the points of intersection for each pair of these lines:Intersection of and :Solving: We find the common solution .Intersection of and :Solving: We find the common solution .Intersection of and :Solving: We find the common solution .ConclusionThe set , representing the union of all intersection points , contains the distinct points , , and . Therefore, there are a total of 3 distinct elements in set .

To determine the set of points (locus) equidistant from specific points, we start by finding the expression for set .Assume a point is equidistant from and . The equation for equidistance between and these two points is: Squaring both sides, we have: Simplifying, we get: Thus, the set is the line , which is the -axis.Next, consider set , the set of points equidistant from (the -axis) and point . This describes a parabola with the following relationship:The distance from the -axis is . Setting this equal to the distance from , we have: Squaring both sides, we derive: Expanding and simplifying gives: For the point , where : Therefore, the value of is 9.