LUANDATO wrote:In how many points does the curve
$$y=x^2−qx+p$$
pass through x-axis?
$$1) q^2>4p$$
$$2) p^2>q$$
The OA is A.
Can any expert help me with this DS question please? I don't have it clear. Thanks.
Since the curve y=x^2−qx+p passes through the X-axis, the Y-coordinates of the points must be 0.
Thus, x^2 − qx + p = 0
=> x = [q ± √(q^2 - 4p)]/2
Case 1: If q^2 = 4p, the curve passes through only one point whose coordinates are (q/2, 0).
Case 2: If q^2 > 4p, the curve passes through two points whose coordinates are ([q + |√(q^2 - 4p)|]/2, 0) & ([q - |√(q^2 - 4p)|]/2, 0).
Case 3: If q^2 < 4p, the curve is imaginary.
Let's see each statement one by one.
(1) q^2 > 4p
This is Case 2. Thus, the curve passes through two points. Sufficient.
(2) p^2 > q
We cannot deduce whether q^2 < = > 4p. Let's see how.
Case 1: Say q = -3, and p = 2, thus q^2 = 9 and 4p = 8. q^2 > 4p. The curve passes through two points.
Case 2: Say q = -3, and p = 4, thus q^2 = 9 and 4p = 16. q^2
< 4p. The curve is imaginary.
Insufficient.
The correct answer:
A
Hope this helps!
-Jay
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