Problem 86 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a \neq 0\) and the line \(2 b x+3 c y+4 d=0\) passes through the points of intersection of the parabolas \(y^{2}=4 a x\) and \(x^{2}=4 a y\), then (A) \(d^{2}+(2 b+3 c)^{2}=0\) (B) \(d^{f}+(3 b+2 c)^{2}=0\) (C) \(d^{2}+(2 b-3 c)^{2}=0\) (D) \(f+(3 b-2 c)^{2}=0\)

Step-by-Step Solution

Answer

The correct answer is (C) \(d^2 + (2b - 3c)^2 = 0\).

Step 1: Identify Intersection Points

To find the points of intersection of the parabolas, solve the system of equations: \[ y^2 = 4ax \] and \[ x^2 = 4ay. \]Setting these equal, we have \[ y^2 = 4a \frac{y^2}{4a} = a \] and \[ x^2 = 4a \frac{x^2}{4a} = a. \]Both expressions equal \(a\), implying \(x = y = 2a\). This gives us the intersection points \((0, 0)\).

Step 2: Plug Intersection Point into Line Equation

Substitute the intersection point \((0,0)\) into the line equation \(2bx + 3cy + 4d = 0\):\[ 2b(0) + 3c(0) + 4d = 0 \Rightarrow 4d = 0 \Rightarrow d = 0. \]

Step 3: Evaluate Each Option

Evaluate each given option using the condition \(d = 0\):- (A) \(d^2 + (2b + 3c)^2 = 0\): This simplifies to \((2b + 3c)^2 = 0\), which implies \(2b + 3c = 0\).- (B) \(d^f + (3b + 2c)^2 = 0\): This is irrelevant as \(f\) is undefined.- (C) \(d^2 + (2b - 3c)^2 = 0\): This simplifies to \((2b - 3c)^2 = 0\), which implies \(2b - 3c = 0\).- (D) \(f + (3b - 2c)^2 = 0\): Like (B), \(f\) is undefined.

Step 4: Conclusion

The correct answer is obtained when we evaluate option C: if \(d = 0\), then the only consistent remaining condition is given by \((2b - 3c)^2 = 0\), leading to \(2b - 3c = 0\). Therefore, the correct answer is (C).

Key Concepts

Parabolas

Parabolas are fundamental shapes in the study of conics. They have distinct properties and appear in real-world situations, like satellite dishes and headlights of cars. A parabola is typically defined by equations of the form \(y^2 = 4ax\) or \(x^2 = 4ay\). These equations describe symmetrical curves related to a point known as the focus and a line called the directrix. The equation \(y^2 = 4ax\) represents a parabola that opens to the right on a coordinate plane. Conversely, \(x^2 = 4ay\) denotes a parabola opening upwards. Both share a vertex at the origin \((0, 0)\) when positioned on a regular coordinate grid. This vertex acts as the starting point or the center of the parabola. Understanding these properties of parabolas helps in solving intersection problems, where you determine common points between different curves.

Intersection Points

Finding the intersection points of conic sections is a crucial step when solving problems that involve multiple curves. In this particular exercise, we focus on two parabola equations, \(y^2 = 4ax\) and \(x^2 = 4ay\). To find their intersection points, you need to solve them simultaneously. The key step is to set both expressions equal and find common solutions. By equating \(y^2 = 4ax\) and \(x^2 = 4ay\), you substitute the expressions to find that both equations simplify to equate their "\(a\)-portion" or constant term. This leads to solving a system of equations, revealing that the intersection point is \((0, 0)\). Understanding how to find intersection points involves recognizing symmetry and balance between the equations. This skill is applicable in numerous mathematical problems and real-world applications like engineering and physics.

Line Equation

In the study of conics, the role of a line equation is often to impose certain conditions or constraints. In this case, we're dealing with a line equation expressed as \(2bx + 3cy + 4d = 0\). To determine the conditions this line imposes, we substitute known intersection points into the equation. For the given problem, the intersection point \((0, 0)\) is plugged into the line equation. Simplifying, you determine that \(4d = 0\), hence, \(d = 0\). This tells us that the line also passes through the origin like the parabolas, thus reinforcing their intersection point. Evaluating the correctness of different algebraic conditions requires substituting parameters and simplifying the expression. By doing this, you can confirm or refute given conditions, like option (C), which meets the required relationship after satisfying \(d = 0\). These procedures widen your understanding of how lines can relate to other shapes in geometry.