TS EAMCET · Maths · Parabola

The locus of a point which divides the line segment joining the focus and any point on the parabola \(y^2=12 x\) in the ratio \(m: n(m+n \neq 0)\) is a parabola. Then the length of the latus rectum of that parabola is

A \(\frac{m}{m+n}\)
B \(\frac{12 m}{m+n}\)
C \(\frac{m}{12(m+n)}\)
D \(\frac{n}{12(m+n)}\)

Verified Solution

Answer & Solution

Correct Answer

(B) \(\frac{12 m}{m+n}\)

Step-by-step Solution

Detailed explanation

Given parabola: \(y^2=12 x \Rightarrow 4a=12 \Rightarrow a=3\). Focus: \(F = (3, 0)\). Parametric point on parabola: \(P = (3t^2, 6t)\). Let the locus point be \((x, y)\). Using section formula for ratio \(m:n\): \(x = \frac{n(3) + m(3t^2)}{m+n}\)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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