Problem 111 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation of the tangent to the circle \(x^{2}+y^{2}+4 x-\) \(4 y+4=0\) which make equal intercepts on the positive co-ordinate axes, is (A) \(x+y=2\) (B) \(x+y=2 \sqrt{2}\) (C) \(x+y=4\) (D) \(x+y=8\)

Step-by-Step Solution

Answer

The correct answer is (B) \(x+y=2\sqrt{2}\).

Step 1: Rewrite the Circle Equation

Start by rewriting the given equation of the circle in the standard form. The given equation is \( x^2 + y^2 + 4x - 4y + 4 = 0 \). To convert to standard form, complete the square for both \(x\) and \(y\).

Step 2: Complete the Square

For \(x\): Take half of 4, square it, and rewrite as \((x + 2)^2 - 4\). For \(y\): Take half of -4, square it, and rewrite as \((y - 2)^2 - 4\). Substitute these back to get: \((x + 2)^2 - 4 + (y - 2)^2 - 4 + 4 = 0\).

Step 3: Simplify the Equation

Combine the constants: \((x+2)^2 + (y-2)^2 = 4\). This represents a circle with center \((-2, 2)\) and radius 2.

Step 4: Equation of Tangent Making Equal Intercepts

The equation of a line making equal intercepts on the axes is of the form \(x + y = c\). For the line to be tangent to the circle, the perpendicular distance from the center to the line must equal the radius.

Step 5: Calculate the Perpendicular Distance

Use the formula for perpendicular distance from a point \((h, k)\) to a line \(Ax + By + C = 0\): \[ \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \] Set \((h, k) = (-2, 2)\) and the line equation \(x + y = c\), i.e., \(A=1, B=1, C=-c\).

Step 6: Set the Distance Equal to Radius

The radius is 2. So, \[ \frac{|1(-2) + 1(2) - c|}{\sqrt{1^2 + 1^2}} = 2 \] This simplifies to \[ \frac{|0 - c|}{\sqrt{2}} = 2 \]. Solve for \(c\).

Step 7: Solve for the Constant

Equate to solve: \( \frac{|c|}{\sqrt{2}} = 2 \)\( |c| = 2\sqrt{2} \) Thus, \( c = 2\sqrt{2} \). This gives the tangent's equation \(x + y = 2\sqrt{2}\).

Key Concepts

Circle Equation

The equation of a circle is a mathematical expression that represents all the points on a plane that are equidistant from a central point, known as the center. The general equation of a circle in its standard form is \[(x - h)^2 + (y - k)^2 = r^2\] where

• \((h, k)\) represents the coordinates of the center of the circle
• \(r\) is the radius of the circle

To transform other forms of circle equations into this standard form, we often use the technique of completing the square, which involves adding and subtracting terms to form perfect square trinomials. For instance, in the original exercise, converting the equation \[x^2 + y^2 + 4x - 4y + 4 = 0\] into the standard form helps us find that the circle's center is at \((-2, 2)\) and its radius is 2. This understanding is crucial for determining further properties of the circle, such as tangent lines.

Perpendicular Distance

The perpendicular distance from a point to a line is a measure of how far the point is from the closest point on the line. This concept is fundamental in coordinate geometry, especially when dealing with tangents to circles. The formula to calculate the perpendicular distance from a point \((h, k)\) to a line given by \(Ax + By + C = 0\) is:\[ \frac{|Ah + Bk + C|}{\sqrt{A^2 + B^2}} \]In the context of the original exercise, to determine if a line is tangent to a circle, this distance should be equal to the radius of the circle. For the exercise in question, you calculate the perpendicular distance from the circle's center \((-2, 2)\) to the line of form \(x + y = c\), ensuring this distance equals the circle's radius, 2.

Line Equation

The equation of a line is a fundamental concept in coordinate geometry, representing all the points along a straight path. A common form is expressed as \(y = mx + c\) where

• \(m\) is the slope of the line
• \(c\) is the y-intercept

Moreover, another form popular in various geometric problems is \(Ax + By + C = 0\), known as the general or standard form. In the problem you explored, the line equation \(x + y = c\) implies that the line makes equal intercepts with the coordinate axes, a key attribute in discerning problems involving symmetry and tangency. Solving for \(c\) leads to understanding when a line touching the circle's circumference precisely fulfills the tangent condition.

Coordinate Geometry

Coordinate geometry, often referred to as analytic geometry, is a branch of geometry where algebraic principles are used to study geometric problems. It involves using a coordinate system to describe the locations of points, lines, and curves, offering a powerful way of solving complex geometric ideas. For instance, in problems involving circles and tangents, coordinate geometry facilitates precise measurements and calculations.

• It allows converting geometric problems into algebraic equations.
• Helps in deriving equations of curves and lines.
• Enables finding intersections and distances between geometric entities.

In the given exercise, coordinate geometry is crucial in translating the problem of a tangent line touching a circle into algebraic equations the aid of concepts like the circle equation, line equation, and perpendicular distance. By mastering these concepts, solving related geometric problems becomes more intuitive and less cumbersome.