6.2 Q-3

Question Statement

The problem asks to determine the position of the point $(5,6)$ with respect to two different circles:

1. For the circle with the equation $x^2 + y^2 = 81$, check the position of the point $(5,6)$.
2. For the circle with the equation $2x^2 + 2y^2 + 12x - 8y + 1 = 0$, check the position of the point $(5,6)$.

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Background and Explanation

To solve this problem, we need to check the position of the point relative to the given circles. The general approach is based on comparing the value of $D$, which is derived by substituting the coordinates of the point into the equation of the circle.

• For a Circle: The position of a point $(x_0, y_0)$ relative to the circle is determined by substituting $(x_0, y_0)$ into the circle’s equation:

$$D = x_0^2 + y_0^2 - r^2$$

where $r$ is the radius of the circle. The position is:

• Inside the circle if $D < 0$,
• On the circle if $D = 0$,
• Outside the circle if $D > 0$.

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Solution

i. Circle $x^2 + y^2 = 81$

Step 1: Substitute the point $(5,6)$ into the equation $x^2 + y^2 = 81$
Substitute $x = 5$ and $y = 6$ into the equation:

$$D = 5^2 + 6^2 - 81$$ $$D = 25 + 36 - 81 = -20$$

Step 2: Interpret the result
Since $D = -20$, which is less than zero, we can conclude that the point $(5,6)$ lies inside the circle.

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ii. Circle $2x^2 + 2y^2 + 12x - 8y + 1 = 0$

Step 1: Simplify the equation
Divide the entire equation by 2 to simplify:

$$x^2 + y^2 + 6x - 4y + \frac{1}{2} = 0$$

Step 2: Substitute the point $(5,6)$ into the simplified equation
Substitute $x = 5$ and $y = 6$ into the equation:

$$(5)^2 + (6)^2 + 6(5) - 4(6) + \frac{1}{2}$$

Simplifying:

$$D = 25 + 36 + 30 - 24 + \frac{1}{2} = 67 + \frac{1}{2} = 67.5$$

Step 3: Interpret the result
Since $D = 67.5$, which is greater than zero, we conclude that the point $(5,6)$ lies outside the circle.

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Key Formulas or Methods Used

• Position of a Point Relative to a Circle:
  The position of a point $(x_0, y_0)$ relative to a circle with equation $x^2 + y^2 = r^2$ is determined by the value of:

$$D = x_0^2 + y_0^2 - r^2$$

• If $D < 0$, the point is inside the circle.
• If $D = 0$, the point is on the circle.
• If $D > 0$, the point is outside the circle.

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Summary of Steps

1. For each circle, substitute the coordinates of the point into the circle’s equation.
2. Calculate the value of $D$.
3. Determine the position of the point based on the value of $D$:
   • If $D < 0$, the point is inside.
   • If $D = 0$, the point is on the circle.
   • If $D > 0$, the point is outside.