The focus of the parabola is the centre of the circle of radius 5 . If the values of , for which C p

Q: JEE Main 2023 Conic Sections: The focus of the parabola is the centre of the circle of radius 5 . If the values of , for which C p...

The correct answer is 2. This problem is a comprehensive test of analytical geometry, requiring a strong grasp of parabolas, circles, and straight lines. The solution involves a sequence of steps: first, identifying the key properties of the parabola to find the center of the circle; second, defining the circle's equation; third, finding the intersection point of two lines in terms of a parameter ; and finally, using the condition that this point lies on the circle to solve for . --- **1. Key Concepts and Formulas** * *

Question

The focus of the parabola is the centre of the circle of radius 5 . If the values of , for which C passes through the point of intersection of the lines and , are and , then is equal to .

Accepted Answer

This problem is a comprehensive test of analytical geometry, requiring a strong grasp of parabolas, circles, and straight lines. The solution involves a sequence of steps: first, identifying the key properties of the parabola to find the center of the circle; second, defining the circle's equation; third, finding the intersection point of two lines in terms of a parameter $\lambda$ ; and finally, using the condition that this point lies on the circle to solve for $\lambda$ .

1. Key Concepts and Formulas

Parabola: The standard form of a parabola opening to the right with its vertex at the origin is $y^2 = 4ax$ . Its focus is at $(a,0)$ . If the parabola is shifted, its equation becomes $(y-k)^2 =

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Solution

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