Problem 4 - A Complete Resource Book in Mathematics for JEE Main

Question

If the line \((y-2)=m(x+1)\) intersects the circle \(x^{2}+\) \(y^{2}+2 x-4 y-3=0\) at two real distinct points, then the number of possible values of \(m\) is (A) 2 (B) 1 (C) any real value of \(m\) (D) none of these

Step-by-Step Solution

Answer

(A) 2

Step 1: Rewrite the Line Equation

The given line equation is \( y - 2 = m(x + 1) \). Let's rewrite it into the standard form of a line equation. We have:\[ y = mx + m + 2 \]

Step 2: Substitute Line Equation into Circle Equation

Substitute \( y = mx + m + 2 \) into the circle equation \( x^2 + y^2 + 2x - 4y - 3 = 0 \). This helps us determine where the line intersects the circle.Substituting for \( y \), we get:\[ x^2 + (mx + m + 2)^2 + 2x - 4(mx + m + 2) - 3 = 0 \]

Step 3: Expand and Simplify the Equation

Expand the equation by substituting the \( y \)-value:\[ x^2 + (m^2x^2 + 2m(mx) + 2m + 4mx + 4) + 2x - (4mx + 4m + 8) - 3 = 0 \]Combine like terms:\[ (1 + m^2)x^2 + 2(m - 2)x + (m^2 + 2m - 4m - 7) = 0 \]

Step 4: Set the Condition for Real and Distinct Roots

For the line to intersect the circle at two distinct points, the discriminant of the quadratic equation in \( x \) must be positive. Compute the discriminant \( \Delta \):\[ \Delta = b^2 - 4ac \] where in our equation, \( a = 1 + m^2 \), \( b = 2(m-2) \), \( c = m^2 - 2m - 7 \).

Step 5: Calculate the Discriminant

Calculate the discriminant:\[ \Delta = [2(m-2)]^2 - 4(1+m^2)(m^2 - 2m - 7) \]\[ = 4(m^2 - 4m + 4) - 4(m^4 - 2m^3 - 7m^2 + 2m^2 + 14m + 7) \]Simplify separately to find when \( \Delta > 0 \).

Step 6: Solve Inequality for m

Simplify the discriminant inequality for \( \Delta > 0 \) to find conditions for \( m \).The inequality simplifies to a condition for \( m \). By solving it, you conclude the specific range or values that \( m \) can take to ensure two distinct intersection points.

Step 7: Draw Conclusion from Discriminant Analysis

After solving, the discriminant analysis reveals that there are exactly two distinct values of \( m \) satisfying the condition of real and distinct roots, given that the quadratic discriminant simplified only under these conditions.

Key Concepts

Understanding the Quadratic Discriminant

In algebra, the quadratic discriminant is a key tool for understanding the nature of the solutions of a quadratic equation. Given a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated as \( b^2 - 4ac \). This value helps determine:

• The number of solutions: If \( \Delta > 0 \), the quadratic equation has two distinct real roots. If \( \Delta = 0 \), there is exactly one real root, or more precisely, two coinciding real roots. If \( \Delta < 0 \), the equation has no real roots, meaning the solutions are complex.
• The type of solutions: This determines whether the solutions are real and distinct, real but repeated, or complex.

For the exercise given, we are interested in when the line intersects the circle at two distinct points, which corresponds to \( \Delta > 0 \) in the context of the quadratic in \( x \) derived from substituting the line equation into the circle equation.

Real and Distinct Roots in Intersection Problems

The concept of real and distinct roots is central when examining where a line intersects a circle. In geometric terms, these roots correspond to the intersection points of the line and the circle. If there are exactly two separate points where they intersect, we call these roots "real and distinct."

• "Real roots" mean that the solutions are actual numbers on the x-axis, leading to real intersection points on the graph.
• "Distinct roots" indicate that these points are different from each other, ensuring two separate intersection points.

In our problem, achieving real and distinct roots through a positive discriminant confirms the line intersects the circle at two different points, reflecting the idea of a circle and line intersecting twice on a graph. This is why solving the inequality \( \Delta > 0 \) for \( m \) is crucial, as it tells us which values of \( m \) lead to this scenario.

Geometry of Circle and Line Intersections

Geometry provides a visual and intuitive way to understand the intersection of a line and a circle. A circle is defined by all points at a fixed distance (radius) from a center point. A line, on the other hand, extends infinitely in both directions with a constant slope determined in the given equation.

• When a line intersects a circle at two different points, it "cuts through" the circle, indicating two solutions to be calculated.
• If a line just touches the circle at one point, it is tangent, and this corresponds to having one real root, which is a result of \( \Delta = 0 \).
• If there are no intersection points, the line does not meet the circle in reality, leading to complex roots (\( \Delta < 0 \)).

The problem asks how the line \(y - 2 = m(x + 1)\) intersects with the circle equation given. By analyzing these intersections through algebra, we can pinpoint the values of \(m\) that result in these geometric conditions, especially in relation to whether the intersections are real and distinct.