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ISI2015-MMA-48

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Suppose the circle with equation $x^2+y^2+2fx+2gy+c=0$ cuts the parabola $y^2=4ax, \: (a>0)$ at four distinct points. If $d$ denotes the sum of the ordinates of these four points, then the set of possible values of $d$ is

  1. $\{0\}$
  2. $(-4a,4a)$
  3. $(-a,a)$
  4. $(- \infty, \infty)$
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Let ... x² + y² + 2gx + 2fy + c = 0 ... (1)

and ... y² = 4ax, a > 0 .................... (2)

From (2), ... x = y² / (4a).

Putting this value of x in (1), we get

( y⁴ / 16a²) + y² + 2g(y² / 2a) + 2fy + c = 0

∴ y⁴ + 0y³ + (16a²+8ag)y² + (32a²f)y + 16a²c = 0 ......... (3)

This equation gives the Ordinates of the 4 points of intersection

of the circle and the parabola.

Sum of These Ordinates

= Sum of Roots of eq(3)

= (-1)* { ( coeff. of y³ ) / ( coeff. of y⁴ )}  [using Vieta's Relation]

= - ( 0 / 1 )

= 0.
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