For the parabola, y 2 = 4px find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

Question

Philoid · Accepted Answer

Let AB be the double ordinate of length 8p for the parabola y² = 4px.

Comparing with standard form we get y² = 4ax we get,

⇒ axis is y = 0

⇒ vertex is O(0, 0).

We know that double ordinate is perpendicular to the axis.

Let us assume that the point at which the double ordinate meets the axis is (x₁, 0).

Then the equation of the double ordinate is y = x₁. It meets the parabola at the points (x₁, 4p) and (x₁, - 4p) as its length is 8p.

Let us find the value of x₁ by substituting in the parabola.

⇒ (4p)² = 4p(x₁)

⇒ x₁ = 4p.

The extremities of the double ordinate are A(4p, 4p) and B(4p, - 4p).

Let us find assume the slopes of OA and OB be m₁ and m₂. Let us find their values.

⇒

⇒ m₁ = 1

⇒

⇒ m₂ = - 1

⇒ m₁.m₂ = 1. - 1

⇒ m₁.m₂ = - 1

We have got product of slopes - 1. So, the lines OA and OB are perpendicular to each other.

So, the extremities of double ordinate make right angle with vertex.

For the parabola, y2 = 4px find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

For the parabola, y² = 4px find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.