Problem 72 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The circle \(x^{2}+y^{2}=4 x+8 y+5\) intersects the line \(3 x-4 y=m\) at two distinct points if (a) \(-35

Step-by-Step Solution

Answer

(a) The circle intersects the line at two distinct points if \(-35 < m < 15\).

Step 1: Rearrange Equation of Circle

The given circle equation is \( x^2 + y^2 = 4x + 8y + 5 \). To express it in standard form, we need to rearrange terms and complete the square for both \( x \) and \( y \). Begin by rewriting the equation as \( x^2 - 4x + y^2 - 8y = 5 \).

Step 2: Complete the Square

To complete the square for the \( x \) terms: \( x^2 - 4x \) becomes \((x - 2)^2 - 4\). For the \( y \) terms: \( y^2 - 8y \) becomes \((y - 4)^2 - 16\). Substitute these into the equation: \((x - 2)^2 - 4 + (y - 4)^2 - 16 = 5\).

Step 3: Simplify the Circle Equation

Combine constant terms: \((x - 2)^2 + (y - 4)^2 - 20 = 5\). This simplifies to: \((x - 2)^2 + (y - 4)^2 = 25\). Now it is in the form of a standard circle equation centered at \((2, 4)\) with radius \(5\).

Step 4: Intersection Condition with Line

For the circle to intersect the line \( 3x - 4y = m \) at two distinct points, the distance \( d \) from the center \((2, 4)\) to the line must be less than the radius \( 5 \).

Step 5: Calculate the Distance from Center to Line

Use the distance formula \( d = \frac{|3 imes 2 - 4 imes 4 - m|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 - 16 - m|}{5} \). Simplify to \( \frac{|10 + m|}{5} \).

Step 6: Solve for Intersection Condition

Set the condition \( \frac{|10 + m|}{5} < 5 \). This gives \(|10 + m| < 25\).

Step 7: Solve the Absolute Value Inequality

The inequality \(|10 + m| < 25\) results in two cases: \(10 + m < 25\) and \(10 + m > -25\). Solving these inequalities, we find \(-35 < m < 15\).

Step 8: Verify Solution Range

The range \(-35 < m < 15\) matches option (a).

Key Concepts

Understanding the Circle Equation

The equation of a circle can be expressed in different forms. One common form is the general form, which is given by \[ x^2 + y^2 = 4x + 8y + 5. \]This may not seem intuitive, so we convert it into the standard form. In the standard form, the circle equation appears as \[ (x - h)^2 + (y - k)^2 = r^2, \]where \((h, k)\) is the circle's center and \(r\) is the radius. This helps in easily identifying the circle's center and radius. To convert to this form, we "complete the square" for both \(x\) and \(y\) terms within the given equation. By rewriting, we found that the circle is centered at \((2, 4)\) with a radius of \(5\). This standard form makes geometrical analysis intuitive and simplifies solving intersection problems.

The Concept of Line Intersection

To determine where and how a line intersects a circle, understanding their relationship geometrically is crucial. In coordinate geometry, an intersection refers to points where the line meets the circle. For two distinct intersection points, the distance from the circle's center to the line must be less than the circle's radius. This guarantees that the line crosses into the circle, reaching two points inside. To find this distance, we use the formula for the distance between a point and a line, \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}, \]where \((x_1, y_1)\) is the point (i.e., the circle's center), and \(Ax + By + C = 0\) is the line's equation. This basics allow us to determine when and if a line intersects a circle and whether it does so at one or multiple points.

Applying Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to answer geometric problems about points, lines, and other figures on a plane. This exercise's main stage involves expressing where a circle and a line on a coordinate plane meet. The emphasis is on using algebraic expressions to evaluate geometric situations:

• The circle's equation was expressed in a form to easily identify its attributes like center and radius.
• By framing the line's position in terms of linear algebra (standard form), logical deductions on its interactions with other shapes are facilitated.

This ability to translate geometric forms into algebraic terms and vice versa is essential to solving complex geometry problems on coordinate planes, making it simpler to visualize and calculate intersections and distances.

Understanding Inequality Conditions

Inequalities are a powerful way to analyze intersection situations mathematically. In our context, the inequality determines the condition for the line to intersect the circle at two distinct points. By laying down that the distance from the circle's center \((2, 4)\) to the line \(3x - 4y = m\) must be less than the radius \(5\), we constructed the inequality:\[\frac{|10 + m|}{5} < 5.\]This simplifies into two inequalities:

• \(10 + m < 25\)
• \(10 + m > -25\)

Solving, we find the permissible values of \(m\) which is \(-35 < m < 15\). This ensures the line actually intersects the circle rather than just resting at a point of tangency or not intersecting at all. Inequalities allow the problem to be framed algebraically and solved systematically, providing clear conditions for geometric behavior.