Question

Let P be the parabola, whose focus is $ (-2, 1) $ and directrix is $ 2x + y + 2 = 0 $. Then the sum of the ordinates of the points on P, whose abscissa is -2, is:

• A. \( \frac{3}{2} \)
• B. \( \frac{5}{2} \)
• C. \( \frac{1}{4} \)
• D. \( \frac{3}{4} \)

Hint:

For problems involving parabolas, remember that the distance from a point on the parabola to the focus is equal to its distance from the directrix. Use this property to set up equations and solve for the coordinates.

Answer 1

The Correct Option is A

The equation of a parabola is given by the definition: the distance from any point on the parabola to the focus is equal to the perpendicular distance from the point to the directrix. 
Given: 
- Focus \( F = (-2, 1) \) 
- Directrix: \( 2x + y + 2 = 0 \) 
The distance from the point \( P(x_1, y_1) \) on the parabola to the focus is: \[ \text{Distance to focus} = \sqrt{(x_1 + 2)^2 + (y_1 - 1)^2} \] The distance from \( P(x_1, y_1) \) to the directrix \( 2x + y + 2 = 0 \) is: \[ \text{Distance to directrix} = \frac{|2x_1 + y_1 + 2|}{\sqrt{2^2 + 1^2}} = \frac{|2x_1 + y_1 + 2|}{\sqrt{5}} \] For \( x_1 = -2 \), we substitute \( x_1 = -2 \) into both expressions: \[ \sqrt{(-2 + 2)^2 + (y_1 - 1)^2} = \frac{|2(-2) + y_1 + 2|}{\sqrt{5}} \] Simplifying both sides, we solve for \( y_1 \). After solving, we find: \[ y_1 = \frac{3}{2} \] 
Thus, the sum of the ordinates of the points on the parabola is \( \frac{3}{2} \).

Answer 2

Equation of the parabola is given by: \[ (x + 2)^2 + (y - 1)^2 = \left(\frac{2x + y + 2}{\sqrt{5}}\right)^2 \] Multiplying both sides by 5: \[ 5[(x + 2)^2 + (y - 1)^2] = (2x + y + 2)^2 \] Substitute \(x = -2\): \[ 5(y - 1)^2 = (y - 2)^2 \] Expanding both sides: \[ 5(y^2 - 2y + 1) = y^2 - 4y + 4 \] \[ 5y^2 - 10y + 5 = y^2 - 4y + 4 \] \[ 4y^2 - 6y + 1 = 0 \] Sum of roots: \[ y_1 + y_2 = \frac{6}{4} = \frac{3}{2} \] \[ \boxed{y_1 + y_2 = \frac{3}{2}} \]