Question

For \(a>b>c>0\), the distance between \((1,1)\) and the point of intersection of the lines \(a x+b y+c=0\) and \(b x+a y+c=0\) is less than \(2 \sqrt{2}\). Then (A) \(a+b-c>0\) (B) \(a-b+c<0\) (C) \(a-b+c>0\) (D) \(a+b-c<0\)

Short answer

C) a-b+c>0

Step-by-step solution

Find the Point of Intersection

To find the intersection point, we must solve the pair of linear equations: 1. Solve for x from the equation 'a x + b y + c = 0', and 2. Solve for y from the equation 'b x + a y + c = 0'.

Solve the Equations Simultaneously

Add the two equations to eliminate c, yielding (a + b)(x + y) = 0. Since a and b are positive and greater than c, x + y must be 0. Thus, the intersection point has the form (d, -d), for some d.

Calculate the Distance from (1,1)

Using the distance formula, the distance D between (1,1) and (d, -d) is \[D = \sqrt{(d-1)^2 + (-d-1)^2}\]

Evaluate the Inequality with the Distance

Use the distance inequality \[D < 2\sqrt{2}\] to find an inequality involving d.

Determine the Correct Inequality

Substitute d with its value in terms of a, b, and c into the inequality obtained from Step 4 and simplify to determine which of the given answer choices is correct.

Key concepts

Linear Equations Intersection

Understanding the intersection of linear equations is essential in coordinate geometry. It is the point where two lines meet on the coordinate plane. To find this point, we look for the values of the variables that satisfy both equations simultaneously. In the given problem, we have two equations,\( ax + by + c = 0 \) and \( bx + ay + c = 0 \).

To find the intersection, we perform algebraic manipulations such as adding, subtracting, or multiplying the equations to eliminate one of the variables. In our example, when we add the equations, we eliminate \( c \), which leads us to \( (a + b)(x + y) = 0 \). Given that \( a, b > 0 \) by the problem's constraints, the only solution for \( x + y = 0 \) is when each variable is the negative of the other, resulting in the intersection point of \( (d, -d) \).

Distance Formula

The distance formula is a fundamental tool in coordinate geometry used to determine the distance between two points in a plane. The formula is derived from the Pythagorean theorem and is given by:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. In the problem at hand, we used the distance formula to calculate the distance \(D\) from the point \((1,1)\) to the intersection point \((d, -d)\), leading to the equation:
\[ D = \sqrt{(d - 1)^2 + (-d - 1)^2} \]
As the problem states, this distance \(D\) must be less than \(2\sqrt{2}\). This inequality helps us derive further information about the relationship between \(a\), \(b\), and \(c\).

Inequality Evaluation

Inequality evaluation is a process of determining the validity of an inequality under given conditions. In our scenario, we derive an inequality involving the distance \( D \) and the coordinates of the points. After establishing that \( D < 2\sqrt{2} \), we express \( D \) in terms of \( d \), which is related to the coefficients \( a \), \( b \), and \( c \) from our original equations.

By squaring both sides of the inequality and simplifying, we convert the initial inequality concerning \( D \) into a more standard form that can be analyzed to understand the relationships between \( a \), \( b \), and \( c \). It will reveal which of the answer choices (A) through (D) correctly describes this relationship. This process exemplifies how inequalities can help in solving problems that involve geometric constraints.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, marries algebra with geometry by using the coordinate plane to analyze and solve geometric problems. The intersection of lines, the application of the distance formula, as well as the evaluation of inequalities—all of which are present in our JEE Advanced mathematics problem—are vital aspects of coordinate geometry.

This branch of mathematics allows the translation of geometric problems into algebraic forms, making them easier to manipulate and solve. Through the use of coordinates, we can accurately describe geometric figures and the relationships between points, lines, and planes. The concepts combined from our linear equations and distance formula allow us to calculate precise distances and describe shapes analytically, a skill that is not only crucial for solving textbook problems but also for practical applications in various scientific fields.