Question

If \(P\) is a point \((x, y)\) on the line \(y=-3 x\) such that \(P\) and the point \((3,4)\) are on the opposite sides of the line \(3 x-4 y=8\), then which of the following is/are FALSE ? (1) \(y<-\frac{8}{5}\) (2) \(y>-\frac{8}{5}\) (3) \(y>-\frac{11}{5}\) (4) \(y<-\frac{9}{5}\)

Short answer

Option (2) is FALSE.

Step-by-step solution

- Determine the coordinates of point P

Given point P is on the line equation: \(y = -3x\). Thus for any point P, the y-coordinate is determined by its x-coordinate as \(y = -3x\).

- Calculate the side of the line for point P

Given line equation: \(3x - 4y = 8\). Substituting point P's coordinates into this equation: \(3x - 4(-3x) = 3x + 12x = 15x\). Hence if \(15x > 8\), point P is on one side of the line, and if \(15x < 8\), point P is on the opposite side.

- Verify the side of the line for point (3,4)

Substitute coordinates (3,4) into the same line equation: \(3(3) - 4(4) = 9 - 16 = -7\). Since \(-7 < 8\), point (3,4) is on one side, and point P, therefore, must be such that \(15x > 8\). Thus \(x > \frac{8}{15}\).

- Translate the x-Constraint for y

Since y = -3x, substituting the x-constraint \(x > \frac{8}{15}\) yields: \(y < -3 \times \frac{8}{15} = -\frac{24}{15} = -\frac{8}{5}\). Thus point P must have y-coordinate satisfying \(y < -\frac{8}{5}\).

- Assess the given options

Option (1) is TRUE since \(y < -\frac{8}{5}\). Option (2) is FALSE since it contradicts the y-constraint. Option (3) could be true as \(y > -\frac{11}{5}\) might hold in some cases. Option (4) is TRUE since \(y < -\frac{8}{5}\) implies \(y < -\frac{9}{5}\) too.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This allows us to analyze geometrical shapes algebraically and understand their relationships via equations.

The exercise involves a point lying on the line represented by the equation \( y = -3x \). In coordinate geometry, an equation like this represents all the points \((x, y)\) that satisfy it. Here, the y-coordinate for any point P can be directly calculated if the x-coordinate is known.

For example, if \( x = 2 \), then \( y = -3 \times 2 = -6 \). This point \((2, -6)\) would lie on the line \( y = -3x \). By understanding this relationship, we can determine specific points that lie on certain curves or lines, which is crucial for solving this problem.

Inequalities

Inequalities tell us about the relationship between two expressions that may not be equal. In the context of coordinate geometry, inequalities are often used to describe regions on a graph that satisfy certain conditions.

In this problem, we're given the introduction of an inequality through line equations. For instance, considering the line \(3x - 4y = 8\), we can determine which side of this line different points lie by substituting their coordinates into the equation.

If the result is greater than 8, the point is on one side; if it is less than 8, it's on the opposite side. For the point \((3,4)\), substituting into \(3x - 4y \) results in -7, indicating it is on the side where the result is less than 8. Understanding such inequalities helps us determine the regions where specific points or areas lie, which is key to solving the exercise.

Line Intersection

Line intersection refers to the point where two lines meet or cross each other. To find the intersection, we set the equations of the two lines equal to each other and solve for the coordinates of the intersection point.

However, in this exercise, we are instead assessing sides of the given line equations to determine their relationships. By identifying the sides on which points \(P\) and \((3,4)\) lie, we understand the intersections better.

For example, line \(3x - 4y = 8\) helps us analyze if a point satisfies \(15x = 3x + 12x = 15x > 8 \) or \(15x < 8\), impacting where these points lie relative to each other. Knowing where a point intersects or lies relative to a line is crucial in many geometry and algebra problems, helping to visualize and understand geometric relationships clearly.