Question

The equation of the line passing through the point $(-4,\,2)$ and the centre of the circle $2x^2 + 2y^2 - 8y = 7$ is

• A. $x + 3y = 2$
• B. $y = 2$
• C. $x = -4$
• D. $x + y = -2$
• E. $y = -2$

Hint:

Always divide the circle equation by the coefficient of $x^2$ (and $y^2$) before completing the square to find the centre. The centre is $(-g,\,-f)$ in the general form $x^2+y^2+2gx+2fy+c=0$.

Answer 1

The Correct Option is B

Step 1: Concept:
• Find the centre of the given circle.
• Then determine the equation of the line passing through the given point and the centre.

Step 2: Find Centre of Circle:
• Given equation: \[ 2x^2 + 2y^2 - 8y = 7 \]
• Divide by 2: \[ x^2 + y^2 - 4y = \frac{7}{2} \]
• Complete the square: \[ x^2 + (y - 2)^2 = \frac{7}{2} + 4 = \frac{15}{2} \]
• So, centre of the circle is: \[ (0,\,2) \]

Step 3: Equation of Line:
• Given point: \((-4,\,2)\)
• Line passes through: \[ (-4,\,2) \text{ and } (0,\,2) \]
• Since both points have same \(y\)-coordinate: \[ \text{Slope} = 0 \]
• Hence, equation of line: \[ y = 2 \]

Step 4: Final Answer:
• \[ y = 2 \]