JEE Main: Sum of Ordinates for Given Abscissa (Parabola Concept) 💡
❓ Question Let P P be the parabola whose focus is ( − 2 , 1 ) (-2,\,1) and whose directrix is 2 x + y + 2 = 0. 2x + y + 2 = 0. Find the sum of the ordinates of the points on P P P whose abscissa is x = − 2. 🖼️ Question Image ✍️ Short Solution A parabola is defined as the locus of a point whose distance from the focus equals its perpendicular distance from the directrix . We will: 1️⃣ Write the focus–directrix condition 2️⃣ Substitute x = − 2 x = -2 3️⃣ Solve for y y 4️⃣ Add the ordinates 🔹 Step 1 — Use focus–directrix definition Let ( x , y ) (x,y) be any point on the parabola. Distance from focus ( − 2 , 1 ) (-2,1) : ( x + 2 ) 2 + ( y − 1 ) 2 \sqrt{(x+2)^2 + (y-1)^2} Distance from directrix 2 x + y + 2 = 0 2x + y + 2 = 0 : ∣ 2 x + y + 2 ∣ 2 2 + 1 2 = ∣ 2 x + y + 2 ∣ 5 \frac{|2x + y + 2|}{\sqrt{2^2 + 1^2}} = \frac{|2x + y + 2|}{\sqrt{5}} Equating squares: ( x + 2 ) 2 + ( y − 1 ) 2 = ( 2 x + y + 2 ) 2 5 (x+2)^2 + (y-1)^2 = \frac{...