JEE Advanced · Mathematics · 13. Parabola

Paragraph:
Consider the circle \(x^2+y^2=9\) and the parabola \(y^2=8 x\). They intersect at \(P\) and \(Q\) in the first and the fourth quadrants, respectively. Tangents to the circle at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(R\) and tangents to the parabola at \(P\) and \(Q\) intersect the \(\mathrm{X}\)-axis at \(S\).Question:
The ratio of the areas of the \(\triangle P Q S\) and \(\triangle P Q R\) is

A
\(1: \sqrt{2}\)
B
\(1: 2\)
C
\(1: 4\)
D
\(1: 8\)

Verified Solution

Answer & Solution

Correct Answer

(C)
\(1: 4\)

Step-by-step Solution

Detailed explanation

Coordinates of \(P\) and \(Q\) are \((1,2 \sqrt{2})\) and \((1,-2 \sqrt{2})\).

Area of \(\triangle P Q R=\frac{1}{2} \cdot 4 \sqrt{2} \cdot 8=16 \sqrt{2}\)
Area of \(\triangle P Q S=\frac{1}{2} \cdot 4 \sqrt{2} \cdot 2=4 \sqrt{2}\)
Ratio of area of \(\triangle P Q S\) and \(\triangle P Q R\) is \(1: 4\).

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.

Same subject

Explore more questions on app

From JEE Advanced

Explore more questions on app