Let PQ be a focal chord of the parabola y2 = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a

Engineering

Mathematics

Tangent of Parabola

Question

Let PQ be a focal chord of the parabola y² = 4ax. The tangents to the parabola at P and Q meet at a point lying on the line y = 2x + a, a > 0.

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Linked Question 1

Length of chord PQ is

Here, t₁t₂ = – 1

and t₁ + t₂ = – 1

(i) Here θ = obtuse angle

$∴$ tan θ < 0

Now, | tan θ | = $| \frac{\frac{2}{t_{1}} - \frac{2}{t_{1}}}{1 + \frac{4}{t_{1} t_{2}}} | = \frac{2}{3} \sqrt{5} \Rightarrow tanθ = \frac{- 2}{3} \sqrt{5}$

(ii) Length PQ

= (a + at₁²) + (a + at₂²)

= a[t₁² + t₂² + 2]

= a[(t₁ + t₂)² – 2t₁t² + 2]

$∴$ Length PQ = 5a

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Linked Question 2

If chord PQ subtends an angle θ at the vertex of y² = 4ax, then tan θ =

$\frac{- 2 \sqrt{7}}{3}$

$\frac{2 \sqrt{5}}{3}$

$\frac{- 2 \sqrt{5}}{3}$

$\frac{2 \sqrt{7}}{3}$