The number of integral values of k for which the line, 3x + 4y = k intersects the circle, x² + y² -

Q: Why can't I just substitute $y = \frac{k - 3x}{4}$ into the circle equation and check the discriminant?

You absolutely could! That is a valid algebraic path. However, it leads to a quadratic equation in $x$ with a discriminant $D > 0$. While mathematically sound, it is computationally heavy and prone to arithmetic errors. The geometric approach using the perpendicular distance $d < r$ is elegant, faster, and gives you a deeper intuition of the circle's spatial relationship with the line.

Q: Why do we use $d < r$ and not $d \le r$?

Great catch! The problem specifies 'two distinct points.' If $d = r$, the line is tangent to the circle, touching it at exactly one point. If $d > r$, the line misses the circle entirely. Only when $d < r$ does the line slice through the interior, creating two distinct intersection points.

Q: When I solved $|11 - k| < 5$, I got confused with the inequality signs. Why does the sign flip?

When you have $-16 < -k < -6$ and you multiply by $-1$ to isolate $k$, you are essentially reflecting the number line. Since $-16$ is to the left of $-6$, multiplying by $-1$ makes $16$ to the right of $6$. Thus, the inequality signs must reverse to maintain the truth of the statement. Always remember: multiplying or dividing an inequality by a negative number flips the direction!

Question

The number of integral values of $k$ for which the line, $3x + 4y = k$ intersects the circle, $x^2 + y^2 - 2x - 4y + 4 = 0$ at two distinct points is

Sigma Percentile · Accepted Answer

### Analyzing the Setup Welcome, fellow explorer of the mathematical universe. Today, we are not just solving a problem; we are visualizing a dance between a line and a circle. Imagine you are standing on a coordinate plane. You have a circle, defined by the equation $x^2 + y^2 - 2x - 4y + 4 = 0$, and a line, $3x + 4y = k$, moving across the plane as we vary $k$. Our goal is to find how many integer values of $k$ allow this line to slice through the circle at two distinct points. Let's begin. ### Unmasking the Circle Before we can analyze the intersection, we must understand our circle. The given equation is in a general form, which is like a locked chest. To see what is inside, we need to complete the square. We group our $x$ and $y$ terms: $$(x^2 - 2x) + (y^2 - 4y) = -4$$ Now, we add the magic constants to complete the squares. For the $x$ terms, we take half of $-2$, which is $-1$, and square it to get $1$. For the $y$ terms, we take half of $-4$, which is $-2$, and square it to get $4$. To keep our equation balanced, we must add these to both sides: $$(x^2 - 2x + 1) + (y^2 - 4y + 4) = -4 + 1 + 4$$ This simplifies beautifully to $(x - 1)^2 + (y - 2)^2 = 1$. Now, the circle reveals its soul: its center $C$ is at $(1, 2)$ and its radius $r$ is $\sqrt{1} = 1$. We have our anchor point! ### The Geometric Dance Now, consider the line $3x + 4y = k$, or $3x + 4y - k = 0$. For this line to intersect the circle at two distinct points, it must pass through the interior of the circle. Geometrically, this means the perpendicular distance $d$ from the center $(1, 2)$ to the line must be strictly less than the radius $r$. If $d = r$, the line is a tangent (one point). If $d > r$, the line is far away (no points). So, our condition is $d < 1$. ### The Inequality Battle Let's calculate that distance $d$ using the standard formula $d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$. Plugging in our values ($A=3, B=4, C=-k, x_0=1, y_0=2$): $$d = \frac{|3(1) + 4(2) - k|}{\sqrt{3^2 + 4^2}} = \frac{|3 + 8 - k|}{\sqrt{9 + 16}} = \frac{|11 - k|}{5}$$ Applying our condition $d < 1$, we get: $$\frac{|11 - k|}{5} < 1 \implies |11 - k| < 5$$ This absolute value inequality tells us that the expression $11 - k$ must be trapped between $-5$ and $5$: $$-5 < 11 - k < 5$$ Subtracting $11$ from all parts, we get $-16 < -k < -6$. Now, multiply by $-1$ and remember to flip those inequality signs! We arrive at $16 > k > 6$, or $6 < k < 16$. ### Final Calculation We are looking for integral values of $k$ in the range $(6, 16)$. These are $k \in \{7, 8, 9, 10, 11, 12, 13, 14, 15\}$. Counting these, we find there are exactly **9** such values. You have successfully navigated the geometry, the algebra, and the logic. Keep this clarity with you; it is the key to mastering JEE Advanced.

Animated Solution for Mathematics - Circles: The number of integral values of $k$ for which the line, $3 x + 4 y = k$ intersects the circle, $x^{2} + y^{2} - 2 x - 4 y + 4 = 0$ at two distinct points is

Enter Numerical Value:

Visualized Solution

The Given Circle Equation

Completing the Square

Center $C$ and Radius $r$

The Intersecting Line $3 x + 4 y = k$

Geometric Condition $d < r$

Perpendicular Distance Formula

Substituting the Values

Simplifying the Distance $d$

Setting Up the Inequality

Solving the Absolute Value

Isolating $k$

Final Range of $k$

Counting Integral Values

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Setup

Unmasking the Circle

The Geometric Dance

The Inequality Battle

Final Calculation

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