MHT CET · Maths · Parabola

The equations of the normals at the ends of the latusrectum of the parabola \(y^{2}=4 a x\) are given by

A \(x^{2}-y^{2}-6 a x+9 a^{2}=0\)
B \(x^{2}-y^{2}-6 a x-6 a y+9 a^{2}=0\)
C \(x^{2}-y^{2}-6 a y+9 a^{2}=0\)
D None of the above

Verified Solution

Answer & Solution

Correct Answer

(A) \(x^{2}-y^{2}-6 a x+9 a^{2}=0\)

Step-by-step Solution

Detailed explanation

The coordinates of the ends of the latusrectum of the parabola \(y^{2}=4 a x\) are \((a, 2 a)\) respectively. The equation of the normal at \((a, 2 a)\) to \(y^{2}=4 a x\) is
\(
y-2 a=\frac{-2 a}{2 a}(x-a)
\)
or \(x+y-3 a=0\)
Similarly, the equation of the normal \((a,-2 a)\) is
\(
x-y-3 a=0
\)
The combined equation is
\(
x^{2}-y^{2}-6 a x+9 a^{2}=0
\)

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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