If the length of the chord of the circle, x² + y² = r² (r 0) along the line, y - 2x = 3 is r then

Q: Why is the triangle equilateral? I thought it was just an isosceles triangle.

Great question! Remember that the distance from the center to any point on the circle is the radius $r$. So, the two sides connecting the center to the chord endpoints are both $r$. Since the problem explicitly states the chord length is also $r$, all three sides are equal. That is the definition of an equilateral triangle!

Q: Can I use the Pythagorean theorem instead of the equilateral triangle altitude formula?

Absolutely! You are thinking like a true physicist. If you drop a perpendicular from the center to the chord, it bisects the chord. You get a right-angled triangle with hypotenuse $r$, base $r/2$, and height $d$. Using Pythagoras: $r^2 = d^2 + (r/2)^2$, which simplifies to $d^2 = 3r^2/4$, or $d = \frac{\sqrt{3}}{2}r$. Both methods lead to the same beautiful result.

Q: Why do we need to square both sides at the end?

We square both sides to eliminate the square roots and isolate $r^2$. Since the question asks for $r^2$, this is the most efficient path. It also helps us avoid dealing with the $\sqrt{3}$ and $\sqrt{5}$ terms directly, making the final calculation much cleaner.

Question

If the length of the chord of the circle, $x^2 + y^2 = r^2 (r > 0)$ along the line, $y - 2x = 3$ is $r$ then $r^2$ is equal to :

Options:

(A) $\frac{9}{24}$

(B) $\frac{24}{5}$

(C) $\frac{12}{5}$

(D) $12$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a circle defined by the equation $x^2 + y^2 = r^2$, which is centered at the origin $(0, 0)$ with radius $r$. A line defined by $y - 2x = 3$ intersects this circle to form a chord.

We are explicitly given that the length of this chord is equal to $r$. This geometric constraint provides the foundation for our entire solution.

### The Hidden Equilateral Triangle

Consider the triangle $	riangle CAB$ formed by the center of the circle $C(0, 0)$ and the two endpoints of the chord, $A$ and $B$. The sides $CA$ and $CB$ are radii of the circle, meaning their length is $r$.

Since the chord $AB$ also has length $r$, the triangle $	riangle CAB$ is an **equilateral triangle**. In any equilateral triangle with side length $r$, the altitude $d$ (the perpendicular distance from the center to the chord) is given by:

$$d = \frac{\sqrt{3}}{2}r$$

### The Algebraic Bridge

Next, we express the line $y - 2x = 3$ in the standard form $2x - y + 3 = 0$. The perpendicular distance $d$ from the origin $(0, 0)$ to this line is calculated using the standard distance formula:

$$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

Substituting $a=2$, $b=-1$, $c=3$, and the point $(0, 0)$, we obtain:

$$d = \frac{|2(0) - 1(0) + 3|}{\sqrt{2^2 + (-1)^2}} = \frac{3}{\sqrt{5}}$$

### The Synthesis

We now equate the two expressions for the distance $d$ to solve for $r$:

$$\frac{\sqrt{3}}{2}r = \frac{3}{\sqrt{5}}$$

To find $r^2$, we square both sides of the equation:

$$\left( \frac{\sqrt{3}}{2}r ight)^2 = \left( \frac{3}{\sqrt{5}} ight)^2$$

$$\frac{3}{4}r^2 = \frac{9}{5}$$

Solving for $r^2$, we find:

$$r^2 = \frac{9}{5} 	imes \frac{4}{3} = \frac{12}{5}$$

The final value is **$r^2 = 2.4$** or **$r^2 = \frac{12}{5}$**.

Animated Solution for Mathematics - Circles: If the length of the chord of the circle, $x^{2} + y^{2} = r^{2} (r > 0)$ along the line, $y - 2 x = 3$ is $r$ then $r^{2}$ is equal to :

Select Answer:

Visualized Solution

Identify the Circle $x^{2} + y^{2} = r^{2}$

The Given Chord

Forming the Triangle

Altitude of Equilateral Triangle

Perpendicular Distance Formula

Substituting the Coordinates

Calculating the Distance $d$

Equating the Two Expressions

Squaring Both Sides

Solving for $r^{2}$

Final Answer

The Sigma Insight: Intercepts Made by a Circle

Analyzing the Setup

The Hidden Equilateral Triangle

The Algebraic Bridge

The Synthesis

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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; } If the length of the chord of the circle, x2+y2=r2(r>0) along the line, y−2x=3 is r then r2 is equal to :