Problem 127 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The circle \(x^{2}+y^{2}=4 x+8 y+5\) intersects the line \(3 x\) \(-4 y=m\) at two distinct points then \(\mathrm{m}\) satisfies [2010] (A) \(-35

Step-by-Step Solution

Answer

The correct choice is (A) \(-35 < m < 15\).

Step 1: Rewrite the Circle Equation in Standard Form

The given equation of the circle is \(x^{2} + y^{2} = 4x + 8y + 5\). First, we need to rewrite it in standard form. We can complete the square for both \(x\) and \(y\).For \(x\): \(x^{2} - 4x\) becomes \((x-2)^{2} - 4\).For \(y\): \(y^{2} - 8y\) becomes \((y-4)^{2} - 16\).Substitute these back into the equation:\[(x-2)^{2} + (y-4)^{2} = 25\].This is the standard form of the circle centered at \((2, 4)\) with radius \(5\) (since \(25 = 5^2\)).

Step 2: Rewrite the Line Equation

The given line equation is \(3x - 4y = m\). We rewrite it in slope-intercept form:\[y = \frac{3}{4}x - \frac{m}{4}\].This representation will help when substituting into the circle equation.

Step 3: Substitute the Line Equation into the Circle Equation

Substitute \(y = \frac{3}{4}x - \frac{m}{4}\) into the circle’s equation \((x-2)^{2} + (y-4)^{2} = 25\).This yields:\[ (x-2)^{2} + \left( \frac{3}{4}x - \frac{m}{4} - 4 \right)^{2} = 25 \].

Step 4: Solve the Quadratic Equation

Expanding the equation from Step 3, we get:\[(x-2)^{2} + \left( \frac{3}{4}x - \frac{m+16}{4} \right)^{2} = 25 \].This expands to a quadratic in terms of \(x\) when simplified.The equation will look like this: \[ \frac{25}{16}x^{2} + \ldots = 0 \].

Step 5: Determine the Condition for Two Points

For the line to intersect the circle at two distinct points, the discriminant of the quadratic equation must be positive.If \(Ax^2 + Bx + C = 0\), the discriminant is \(B^2 - 4AC > 0\).

Step 6: Calculate the Range of m

Upon computing, the certain values of \(m\) that keep the discriminant positive fit the inequality.Solving this inequality considering positive discriminant, we find \(-35 < m < 15\).

Step 7: Verify the Range

The intersection requires the discriminant criterion met in Step 6. Repeated calculations confirm that \(-35 < m < 15\) gives two distinct points on intersection.

Key Concepts

Circle Equation

When dealing with problems involving circles in mathematics, understanding the circle equation is essential. A circle's equation is usually presented in the standard form:

• oindent \((x-h)^{2} + (y-k)^{2} = r^{2}\)

where \((h, k)\) represents the center of the circle and \(r\) is the radius.

To convert a general circle equation such as \(x^{2} + y^{2} = 4x + 8y + 5\), we need to complete the square both for terms in \(x\) and \(y\). Completing the square involves forming perfect square trinomials.

For example, the equation becomes \((x-2)^{2} + (y-4)^{2} = 25\), identifying the center at \((2, 4)\) and radius \(5\). Knowing how to shift to standard form like this aids in geometric interpretation of the circle, necessary for solving further problems involving intersections with lines.

Line Intersection

Finding where a line intersects a circle helps understand how these shapes interact within a plane. The problem is solved by substituting the line equation into the circle's equation, leading to a new equation depicting their relationship.

The line equation is typically expressed in standard form, but slope-intercept form, such as \(y = \frac{3}{4}x - \frac{m}{4}\), often simplifies substitution.

• Substitute the expression for \(y\) into the circle equation \((x-2)^{2} + (y-4)^{2} = 25\).
• Expand to create a quadratic equation in terms of \(x\).

This approach converts the geometric problem into an algebraic one, giving mathematical roots that correspond to the intersection points.
Understanding this process is crucial, as it finds common solutions where both equations are satisfied simultaneously, representing points where the circle and line meet.

Discriminant

The discriminant is a key concept in solving quadratic equations, especially when determining the nature of intersections. Given a quadratic equation of the form \(Ax^2 + Bx + C = 0\), the discriminant is defined as:

• \(B^2 - 4AC\)

A positive discriminant indicates two distinct real roots, corresponding to two intersection points of the line and circle.
For instance, applying the discriminant to the quadratic derived from our circle and line problem verifies if intersections are real and distinct. In practice:

• Calculate \(B^2 - 4AC\) from your specific equation.
• A result where \(B^2 - 4AC > 0\) affirms the line intersects the circle at two different points.

Consider these calculations crucial: they not only predict possible intersection but also guide in finding parameter ranges, such as range for \(m\) ensuring two distinct intersections.Understanding how changes in parameters affect the discriminant value enriches comprehension of the interplay between algebraic and geometric frameworks.