AP EAMCET · Maths · Parabola
If the locus of a point that divides a chord of slope 2 of the parabola \(y^2=4 x\) internally in the ratio \(1: 2\) is a parabola, then its vertex is
- A \(\left(\frac{2}{9}, \frac{8}{9}\right)\)
- B \(\left(\frac{1}{9}, \frac{3}{9}\right)\)
- C \(\left(\frac{4}{9}, \frac{8}{9}\right)\)
- D \(\left(\frac{2}{9}, \frac{4}{9}\right)\)
Answer & Solution
Correct Answer
(A) \(\left(\frac{2}{9}, \frac{8}{9}\right)\)
Step-by-step Solution
Detailed explanation
Let the chord endpoints be \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). For \(y^2=4x\), the slope \(m = \frac{y_2-y_1}{x_2-x_1} = \frac{4}{y_1+y_2}\). Given \(m=2\), so \(2 = \frac{4}{y_1+y_2} \implies y_1+y_2=2\). Let the locus point be \(L(h, k)\). It divides PQ in ratio \(1:2\):…
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