The circle x² + y² = 4x + 8y + 5 intersects the line 3x - 4y = m at two distinct points if

Q: Why is it strictly less than ($<$) and not less than or equal to ($\le$)?

The problem asks for two *distinct* points. If the distance equals the radius, the line is a tangent, touching at exactly one point. We need the line to cut through the circle, so the distance must be strictly less than the radius.

Q: Can I use the discriminant method instead?

Absolutely! You can substitute $y = \frac{3x - m}{4}$ into the circle equation to get a quadratic in $x$. For two distinct roots, the discriminant $D$ must be greater than zero. Both methods are elegant, but the geometric approach is often faster.

Q: How do I handle the absolute value inequality $|m + 10| < 25$?

Remember that $|x| < a$ is equivalent to $-a < x < a$. It's a fundamental property of inequalities. So, $|m + 10| < 25$ becomes $-25 < m + 10 < 25$.

Question

The circle $x^2 + y^2 = 4x + 8y + 5$ intersects the line $3x - 4y = m$ at two distinct points if

Options:

(A) $-35 < m < 15$

(B) $15 < m < 65$

(C) $35 < m < 85$

(D) $-85 < m < -35$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a vast, coordinate-mapped plane. Before you lies a mysterious equation: $x^2 + y^2 - 4x - 8y - 5 = 0$.

It looks like a jumble of variables, but to a trained eye, it is a circle waiting to be revealed. This is the first step in our journey: unmasking the geometry.

### The Hidden Identity

To understand this circle, we must bring it into the light of its standard form. We group the $x$ terms and the $y$ terms:

$$(x^2 - 4x) + (y^2 - 8y) = 5$$

Now, we perform the magic of completing the square. For the $x$ terms, we take half of $-4$, which is $-2$, and square it to get $4$. For the $y$ terms, we take half of $-8$, which is $-4$, and square it to get $16$.

To keep the universe in balance, we add these same values to the right side:

$$(x^2 - 4x + 4) + (y^2 - 8y + 16) = 5 + 4 + 16$$

This simplifies beautifully to:

$$(x - 2)^2 + (y - 4)^2 = 25$$

Now, the circle stands revealed: its center $C$ is at $(2, 4)$ and its radius $r$ is $\sqrt{25} = 5$.

### The Geometric Gatekeeper

We are given a line, $3x - 4y = m$, or $3x - 4y - m = 0$. The problem asks for the condition under which this line intersects the circle at two distinct points.

Geometrically, this means the line must be a **secant**. Think of it as a sword slicing through the circle.

For this to happen, the perpendicular distance $d$ from the center of the circle to the line must be strictly less than the radius $r$. If $d = r$, the line is a tangent, touching only once. If $d > r$, the line misses the circle entirely.

Thus, our condition is $d < r$.

### The Bridge of Distance

Now, we build the bridge between geometry and algebra using the perpendicular distance formula:

$$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

Here, our point is $(2, 4)$ and our line is $3x - 4y - m = 0$. Substituting these values, we get:

$$d = \frac{|3(2) - 4(4) - m|}{\sqrt{3^2 + (-4)^2}}$$

Simplifying the numerator, we have $|6 - 16 - m| = |-10 - m| = |m + 10|$. The denominator is $\sqrt{9 + 16} = 5$.

So, the distance is:

$$d = \frac{|m + 10|}{5}$$

### The Final Inequality

We are at the final frontier. We set $d < r$, which means:

$$\frac{|m + 10|}{5} < 5$$

Multiplying both sides by $5$, we get $|m + 10| < 25$. Remember that $|x| < a$ implies the inequality $-a < x < a$.

Therefore, we set up the compound inequality:

$$-25 < m + 10 < 25$$

Subtracting $10$ from all parts, we arrive at the final range:

$$-35 < m < 15$$

### Conclusion

And there it is! The parameter $m$ must live in the open interval **$(-35, 15)$**.

We have taken a complex algebraic equation, transformed it into a clear geometric picture, and solved it with the precision of a master. You have mastered the dance of the line and the circle.

Animated Solution for Mathematics - Circles: The circle $x^{2} + y^{2} = 4 x + 8 y + 5$ intersects the line $3 x - 4 y = m$ at two distinct points if

Select Answer:

Visualized Solution

The Circle's Equation

Completing the Square

Center and Radius

The Secant Line Condition

Setting up the Distance Formula

Substituting the Values

Simplifying the Distance

Applying the Secant Condition

Solving the Absolute Value Inequality

Finding the Range of $m$

Final Conclusion

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

The Hidden Identity

The Geometric Gatekeeper

The Bridge of Distance

The Final Inequality

Conclusion

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Animated Solution for Mathematics - Circles: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The circle x2+y2=4x+8y+5 intersects the line 3x−4y=m at two distinct points if