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Parabola question

2022 · 30 Jun · Shift 1 · Q39
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  5. /2022 · 30 Jun · Shift 1 · Q39

Parabola question

2022 · 30 Jun · Shift 1 · Q39

JEE MainMathematicsParabolaNumerical+4 / −1
Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of Δ\DeltaΔ POQ is equal to ‾\underline{\hspace{2cm}}​.
Numerical answer
View written solutionFree

Correct answer: 25

  1. Parabola and its focus

    Given parabola: y2=4xy^2=4xy2=4x Comparing with the standard form y2=4axy^2=4axy2=4ax, we get a=1a=1a=1 So the focus is S(1,0)S(1,0)S(1,0) and the vertex is O(0,0)O(0,0)O(0,0).

  2. Parametric points of a focal chord

    A general point on the parabola y2=4axy^2=4axy2=4ax is P(at2,2at)P(at^2,2at)P(at2,2at) For a=1a=1a=1, this becomes P(t2,2t)P(t^2,2t)P(t2,2t)

    If PQPQPQ is a focal chord, then the corresponding parameters are ttt and −1t-\frac{1}{t}−t1​.

    Hence, P(t2,2t),Q(1t2,−2t)P(t^2,2t), \qquad Q\left(\frac{1}{t^2},-\frac{2}{t}\right)P(t2,2t),Q(t21​,−t2​)

  3. Use the length of focal chord

    Length of focal chord of parabola y2=4axy^2=4axy2=4ax with endpoints corresponding to ttt and −1t-\frac{1}{t}−t1​ is PQ=a(t+1t)2PQ=a\left(t+\frac{1}{t}\right)^2PQ=a(t+t1​)2

    Here a=1a=1a=1 and PQ=6.25=254PQ=6.25=\frac{25}{4}PQ=6.25=425​, so (t+1t)2=254\left(t+\frac{1}{t}\right)^2=\frac{25}{4}(t+t1​)2=425​

  4. Area of triangle POQPOQPOQ

    Since O=(0,0)O=(0,0)O=(0,0), area of △POQ\triangle POQ△POQ is Area=12∣x1y2−x2y1∣\text{Area} = \frac12 \left|x_1y_2-x_2y_1\right|Area=21​∣x1​y2​−x2​y1​∣

    Substituting P(t2,2t),Q(1t2,−2t)P(t^2,2t), \quad Q\left(\frac{1}{t^2},-\frac{2}{t}\right)P(t2,2t),Q(t21​,−t2​)

    we get Area=12∣t2(−2t)−1t2(2t)∣\text{Area} = \frac12 \left| t^2\left(-\frac{2}{t}\right)-\frac{1}{t^2}(2t)\right|Area=21​​t2(−t2​)−t21​(2t)​ =12∣−2t−2t∣=\frac12 \left|-2t-\frac{2}{t}\right|=21​​−2t−t2​​ =∣t+1t∣=\left|t+\frac{1}{t}\right|=​t+t1​​

  5. Compute the area

    From step 3, (t+1t)2=254\left(t+\frac{1}{t}\right)^2=\frac{25}{4}(t+t1​)2=425​ Therefore, ∣t+1t∣=52\left|t+\frac{1}{t}\right|=\frac{5}{2}​t+t1​​=25​

    So the area is 52\frac{5}{2}25​

  6. Find 10 times the area

    10×Area=10×52=2510 \times \text{Area} = 10 \times \frac{5}{2} = 2510×Area=10×25​=25

  7. Final answer

    25\boxed{25}25​

The derived answer matches the stored correct answer.

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