AP EAMCET · Maths · Parabola

The normal at a point on the parabola $y^{2} = 4 x$ passes through $(5, 0)$ . If two more normals to this parabola also pass through $(5, 0),$ then the centroid of the triangle formed by the feet of these three normals is

A $(\frac{1}{2}, \frac{1}{2})$
B $(2, 0)$
C $(5, 0)$
D $(0, 2)$

Verified Solution

Answer & Solution

Correct Answer

(B) $(2, 0)$

Step-by-step Solution

Detailed explanation

The given equation of a parabola, y2=4x The equation of normal at a point P(5,0) y=-tx+2t+t3 0=-5t+2t+t3 3t=t3 t2=3 t=±3,0 Here, a=1 Co-normal point Pat12,2at1=(3,23) Qat22,2at2=(3,-23) Rat32,2at3=(0,0) The centroid is, C=3+3+03,23-23+03 C=63,0=2,0

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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The normal at a point on the parabola y2=4x passes through (5,0). If two more normals to this parabola also pass through(5,0), then the centroid of the triangle formed by the feet of these three normals is

Step-by-step Solution

The normal at a point on the parabola $y^{2} = 4 x$ passes through $(5, 0)$ . If two more normals to this parabola also pass through $(5, 0),$ then the centroid of the triangle formed by the feet of these three normals is