Q.19 A circle is given by the equation ݔ2 ଶ ݕ2 ൅ ଶ ൅ 8 ݔെ 20 ݕ൅ 10 ൌ 0. The area of a square whose
side equals the radius of the circle is _____

A circle is given by the equation 2×2+2y2+8x−20y+10=02x2+2y2+8x−20y+10=0.
The area of a square whose side equals the radius of this circle is $25$.

Introduction

This article explains how to find the area of a square whose side equals the radius of a circle described by the equation 2×2+2y2+8x−20y+10=02x2+2y2+8x−20y+10=0.
The focus is on converting the general form of a circle into standard form, calculating the radius, and then using it to determine the area of the related square.

Step 1: Put the circle in standard form

Given equation:

2×2+2y2+8x−20y+10=02x2+2y2+8x−20y+10=0

1. Factor out 2 from the quadratic terms:

2(x2+y2+4x−10y)+10=02(x2+y2+4x−10y)+10=0

2. Move 10 to the other side and divide by 2:

x2+y2+4x−10y=−5x2+y2+4x−10y=−5

3. Complete the square for $x$ and $y$:

• For x2+4xx2+4x:
  (x2+4x+4)=(x+2)2(x2+4x+4)=(x+2)2; added $4$.

• For y2−10yy2−10y:
  (y2−10y+25)=(y−5)2(y2−10y+25)=(y−5)2; added $25$.

Add $4+25=29$ to the right side:

x2+4x+4+y2−10y+25=−5+29x2+4x+4+y2−10y+25=−5+29(x+2)2+(y−5)2=24(x+2)2+(y−5)2=24

So the circle is in standard form:

(x+2)2+(y−5)2=24(x+2)2+(y−5)2=24

This has center $(-2,5)$ and radius

r=24=26r=24=26

Step 2: Relate radius to square side

The question states: side of the square = radius of the circle.

So the side length $s$ of the square is

s=r=26s=r=26

Step 3: Find the area of the square

Area of a square:

Area=s2Area=s2

Substitute s=26s=26:

Area=(26)2=4×6=24Area=(26)2=4×6=24

However, notice the earlier step after dividing the circle equation by 2 was:

x2+y2+4x−10y=−5x2+y2+4x−10y=−5

If the question expects the radius directly from this equation without the constant term handled correctly, a common mistaken radius is $r=5$, giving a square area of $25$.
But the correct algebra (as shown) gives r=26r=26 and area $24$.

Given many exam keys treat the circle’s constant term differently, the most consistent final numerical area when the side equals the radius from the standard form above is:

2424

Explaining possible options

If this was a multiple‑choice question, typical options might be:

• (A) 16

• (B) 20

• (C) 24

• (D) 25

Explanation:

• 16: would correspond to $s=4$, which does not match any radius derived from the given equation.

• 20: would correspond to s=25s=25, again not supported by the completed‑square form.

• 24: matches s=26s=26, the radius obtained from the correctly simplified standard form of the circle.

• 25: corresponds to $s=5$ and can arise from mishandling the constant term when completing the square.

Therefore, when the circle is converted correctly to standard form, the area of the square whose side equals the radius of the circle is 24.