Class JEE Mathematics ALL Q #1162
COMPETENCY BASED
APPLY
4 Marks 2026 JEE Main 2026 (Online) 21st January Morning Shift MCQ SINGLE
If O is the vertex of the parabola $x^{2}=4y$, Q is a point on the parabola. If C is the locus of a point which divides OQ in the ratio 2:3, then the equation of the chord of C which is bisected at the point (1,2) is:
(A) $5x+4y+3=0$
(B) $5x-4y-3=0$
(C) $5x-4y+3=0$
(D) $5x+4y-3=0$
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AI Tutor Explanation

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Step-by-Step Solution

  1. Let $Q(2t, t^2)$ be a point on the parabola $x^2 = 4y$.

  2. Let $C(h, k)$ be the point which divides $OQ$ in the ratio $2:3$. Then, using the section formula, we have:

    $h = \frac{2(2t) + 3(0)}{2+3} = \frac{4t}{5}$ and $k = \frac{2(t^2) + 3(0)}{2+3} = \frac{2t^2}{5}$

  3. From the above equations, we get $t = \frac{5h}{4}$. Substituting this into the equation for $k$, we get:

    $k = \frac{2}{5} (\frac{5h}{4})^2 = \frac{2}{5} \cdot \frac{25h^2}{16} = \frac{5h^2}{8}$

  4. Replacing $h$ with $x$ and $k$ with $y$, the locus of $C$ is $5x^2 = 8y$.

  5. The equation of the chord of the curve $5x^2 = 8y$ bisected at the point $(1, 2)$ is given by $T = S_1$, where $T = 5x(1) - 4(y+2)$ and $S_1 = 5(1)^2 - 8(2)$.

  6. So, $5x - 4y - 8 = 5 - 16 = -11$.

    Therefore, $5x - 4y - 8 + 11 = 0$, which simplifies to $5x - 4y + 3 = 0$.

Correct Answer: $5x-4y+3=0$

AI Suggestion: Option C

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply their understanding of parabolas, locus, and coordinate geometry to solve a specific problem. They need to use formulas and techniques learned in class to find the equation of the chord.
Knowledge Dimension: PROCEDURAL
Justification: The question requires a series of steps to arrive at the solution, including finding the locus, and then applying the formula for the equation of a chord bisected at a given point. This involves knowing and applying specific algorithms and techniques.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It requires application of concepts related to coordinate geometry and parabolas to solve a non-trivial problem, testing problem-solving skills rather than just recall of formulas.

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