If the circles x² + y² + 2ax + cy + a = 0 and x² + y² - 3ax + dy - 1 = 0 intersect in two distinct p

Q: Wait, why do we subtract the two circle equations? Can't we just find the intersection points $P$ and $Q$ and then find the line equation?

That is a brilliant question! While you *could* solve for the intersection points, it would be algebraically nightmarish. The beauty of the radical axis (the common chord) is that $S_1 - S_2 = 0$ automatically eliminates the $x^2$ and $y^2$ terms, leaving you with a linear equation. It is a shortcut that saves you from solving a system of non-linear equations!

Q: Why did we set the ratios of the coefficients equal to each other?

When two equations represent the same line, they are essentially the same geometric object. If $L_1: A_1x + B_1y + C_1 = 0$ and $L_2: A_2x + B_2y + C_2 = 0$ are the same line, then $L_1 = kL_2$. This implies that the ratios of their coefficients must be equal: $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$. It is the fundamental way to compare two linear equations.

Q: I got $a^2 + a + 1 = 0$. How do I know if there are real values for $a$?

Great work reaching that quadratic! To check for real roots, we look at the discriminant $D = B^2 - 4AC$. If $D$ is negative, the roots are complex, meaning there are no real values of $a$ that satisfy the condition. In this case, $1^2 - 4(1)(1) = -3$, which is indeed less than zero.

Q: Does the value of $b$ or $c$ or $d$ matter here?

They matter for the existence of the line, but notice that the proportionality condition $\frac{5a}{5} = \frac{a+1}{-a}$ allowed us to isolate $a$ independently. The other variables are constrained by the geometry, but the question asks specifically about the existence of $a$.

Q: What if the circles were tangent? Would the common chord still exist?

If the circles were tangent, the 'common chord' would become the 'common tangent' at the point of contact. The math $S_1 - S_2 = 0$ still holds perfectly! The only difference is that the two intersection points $P$ and $Q$ would coincide.

Question

If the circles $x^2 + y^2 + 2ax + cy + a = 0$ and $x^2 + y^2 - 3ax + dy - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5x + by - a = 0$ passes through $P$ and $Q$ for

Options:

(A) exactly one value of $a$

(B) no value of $a$

(C) infinitely many values of $a$

(D) exactly two values of $a$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing in a vast, empty coordinate plane. Suddenly, two circles appear. One is defined by $x^2 + y^2 + 2ax + cy + a = 0$ and the other by $x^2 + y^2 - 3ax + dy - 1 = 0$.

They are not just sitting there; they are locked in a dance, intersecting at two distinct points, $P$ and $Q$. You are told that a mysterious line, $5x + by - a = 0$, passes exactly through these two points.

Your mission is to determine for how many values of $a$ this is possible. Let us embark on this journey.

### The Secret of the Common Chord

In the world of coordinate geometry, when two circles intersect, the line passing through their intersection points is known as the **common chord** (or the radical axis).

You might be tempted to find the coordinates of $P$ and $Q$ by solving the system of equations, but I urge you to pause. That path is a labyrinth of quadratic roots and frustration.

Instead, we use the elegance of the radical axis theorem. If we have two circles $S_1 = 0$ and $S_2 = 0$, the equation of the line passing through their intersection is simply $S_1 - S_2 = 0$.

Because at the points of intersection, both $S_1$ and $S_2$ are zero, their difference must also be zero. Since the $x^2$ and $y^2$ terms are identical in both equations, they vanish upon subtraction, leaving us with a beautiful, clean linear equation.

### The Algebraic Unveiling

Let us perform the subtraction carefully. We take our first circle $S_1: x^2 + y^2 + 2ax + cy + a = 0$ and our second circle $S_2: x^2 + y^2 - 3ax + dy - 1 = 0$.

Subtracting them gives us:

$$ (x^2 + y^2 + 2ax + cy + a) - (x^2 + y^2 - 3ax + dy - 1) = 0 $$

As expected, the quadratic terms $x^2$ and $y^2$ cancel out, leaving us with:

$$ (2ax - (-3ax)) + (cy - dy) + (a - (-1)) = 0 $$

Simplifying this, we arrive at the equation of our common chord:

$$ 5ax + (c - d)y + (a + 1) = 0 $$

### The Proportionality Test

We are given that the line $5x + by - a = 0$ is the same line as the one we just derived. In the language of linear algebra, if two equations represent the same line, their coefficients must be proportional.

We set up our ratios:

$$ \frac{5a}{5} = \frac{c - d}{b} = \frac{a + 1}{-a} $$

Focusing on the first and the last parts of this equality, we isolate the variable $a$:

$$ a = \frac{a + 1}{-a} $$

Cross-multiplying gives us the quadratic equation:

$$ -a^2 = a + 1 \Rightarrow a^2 + a + 1 = 0 $$

### The Final Verdict

We have arrived at $a^2 + a + 1 = 0$. To determine if there are any real values for $a$, we look at the discriminant $D = B^2 - 4AC$.

Here, $A=1, B=1, C=1$. Calculating this, we get:

$$ D = (1)^2 - 4(1)(1) = 1 - 4 = -3 $$

Since the discriminant is negative, there are no real roots for $a$. The quadratic equation $a^2 + a + 1 = 0$ only yields complex solutions.

Therefore, there is **no real value of $a$** for which the given line can be the common chord of these two circles. We have navigated the geometry, mastered the algebra, and uncovered the truth: the condition is impossible for any real $a$.

Animated Solution for Mathematics - Circles: If the circles $x^{2} + y^{2} + 2 a x + cy + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ intersect in two distinct points $P$ and $Q$ then the line $5 x + b y - a = 0$ passes through $P$ and $Q$ for

Select Answer:

Visualized Solution

Visualizing the Intersection

The Common Chord

Equation of Common Chord

Setting up the Subtraction

Performing the Subtraction

Simplifying the Derived Chord

Comparing with the Given Line

Setting up the Proportionality

Isolating the Variable $a$

Forming the Quadratic Equation

Analyzing the Roots

Calculating the Discriminant

Final Conclusion

The Sigma Insight: Director Circle and Family of Circles

Analyzing the Setup

The Secret of the Common Chord

The Algebraic Unveiling

The Proportionality Test

The Final Verdict

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