Consider the circle and the parabola . If the set of all values of , for which three chords of the c

Q: JEE Main 2023 Conic Sections: Consider the circle and the parabola . If the set of all values of , for which three chords of the c...

The correct answer is 0. This problem involves properties of circles, parabolas, and the concept of chords and their midpoints. We are looking for values of such that there are exactly three distinct chords of a given circle, all passing through the point , and all bisected by a given parabola. --- **1. Key Concept: Equation of a Chord Given its Midpoint** For a conic section given by the general equation , if a point is the midpoint of a chord, then the equation of that chord is given by the formula . Let's break down

Question

Consider the circle and the parabola . If the set of all values of , for which three chords of the circle on three distinct lines passing through the point are bisected by the parabola is the interval , then is equal to .

Accepted Answer

This problem involves properties of circles, parabolas, and the concept of chords and their midpoints. We are looking for values of $\alpha$ such that there are exactly three distinct chords of a given circle, all passing through the point $(\alpha, 0)$ , and all bisected by a given parabola.

1. Key Concept: Equation of a Chord Given its Midpoint

For a conic section given by the general equation $S=0$ , if a point $M(x_1, y_1)$ is the midpoint of a chord, then the equation of that chord is given by the formula $T=S_1$ .

Let's break down this formula for our specific circle $C: x^2+y^2=4$ :

The equation

Question

Solution

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