Parametrics (1 Viewer)

[phoenix159]

The normals to the parabola x² = 4Ay at the points P₁ and P₂ intersect at Q. If the chord P₁P₂ varies in such a way that it always passes through the point (0, –2A), show that Q lies on the parabola.

 

[HeroicPandas]

Step 1: Read question
Step 2: Draw diagram
Step 3: Read question again
Step 4: Identify an aim
Step 5: Work towards your aim by using information given

 

[hayabusaboston]

Step 1: Read question
Step 2: Draw diagram
Step 3: Read question again
Step 4: Identify an aim
Step 5: Work towards your aim by using information given

ahh shit I cant rep u anymore lol.

 

[HeroicPandas]

What is your aim Mr Pheonix?! Show that Q lies on the parabola right? That means Q must satisfy the parabola right?(keep this in your mind)

What is Q? It is the intersection between the normals at P₁ and P₂. Find the point of intersection of these normals to find Q

If we sub in Q into the parabola, we'll need to show that LHS = RHS to show that it satisfies it --> We are not going to sub in Q yet as it looks very BAD --> Q( -ap₁p₂(p₁ + p₂), 2a + a(p₁² + p₁q₂ + p₂²)

Read the question again, notice that it says the line P1P2 must intersect (0,-2a) --> This means (0,-2a) satisfies the EQUATION P1P2

Workout equation P1P2, sub in (0,-2a) to get a relationship between p₁, p₁, p₂ , q₁ and q₂ --> Use this to manipulate the point Q and then we can proceed to subbing it into the parabola to show it satisfies