Home
Class 12
MATHS
If the normals to the parabola y^2=4a x ...

If the normals to the parabola `y^2=4a x` at three points `(a p^2,2a p),` `(a q^2,2a q)` and `(a r^2,2a r)` are concurrent, then the common root of equations `P x^2+q x+r=0` and `a(b-c)x^2+b(c-a)x+c(a-b)=0` is

A

A. p

B

B. q

C

C. r

D

D. 1

Text Solution

Verified by Experts

The correct Answer is:
D
(4) Normal at points `(ap^(2),2ap),(aq^(2),2aq),and(ar^(2),2ar)` are concurrent.
Hence, the points are co-normal points. Therefore,
p+qr=0
So, `px^(2)+qx+r=0` has one root which is x=1.
Therefore, the common root is 1, which also satisfies the second equation.
Promotional Banner

Topper's Solved these Questions

  • PARABOLA

    CENGAGE PUBLICATION|Exercise EXERCISE (MULTIPLE CORRECT ANSWER TYPE )|26 Videos

  • PARABOLA

    CENGAGE PUBLICATION|Exercise LINKED COMPREHENSION TYPE|45 Videos

  • PARABOLA

    CENGAGE PUBLICATION|Exercise Concept Applications Exercise 5.7|9 Videos

  • PAIR OF STRAIGHT LINES

    CENGAGE PUBLICATION|Exercise Numberical Value Type|5 Videos

  • PERMUTATION AND COMBINATION

    CENGAGE PUBLICATION|Exercise Comprehension|8 Videos

Similar Questions

Explore conceptually related problems

If p,q are odd integers, then the roots of the equation 2px^(2)+(2p+q)x+q=0 are

If p and q are the roots of the equation x^2+px+q =0, then

If a ,b ,c in R^+and 2b=a+c , then check the nature of roots of equation a x^2+2b x+c=0.

If PQ and Rs are normal chords of the parabola y^(2) = 8x and the points P,Q,R,S are concyclic, then

If p. q are odd integers. Then the roots of the equation 2px^(2)+(2p+q)x+q=0 are

If , p , q are the roots of the equation x^(2)+px+q=0 , then

If a ,b ,p ,q are non zero real numbers, then how many comman roots would two equations: 2a^2x^2-2a b x+b^2=0 and p^2x^2+2p q x+q^2=0 have?

Two roots of the equation, x^3+qx^2+11x-p =0 are 2 and 3, find (p-q)

The circle x^2+y^2=5 meets the parabola y^2=4x at P and Q . Then the length P Q is equal to (a)2 (b) 2sqrt(2) (c) 4 (d) none of these

If the equation x^2+2x+3=0 and ax^2+bx+c=0 , a, b, c in R, have a common root, then a:b:c is

CENGAGE PUBLICATION-PARABOLA-EXERCISE (SINGLE CORRECT ANSWER TYPE )
  1. If line P Q , where equation is y=2x+k , is a normal to the parabola w...

    Text Solution

    |

  2. Evaluate, min[(x1-x2)^2+(5+sqrt(1-x1^2)-sqrt(4x2))^2],AAx1,x2 in R , ...

    Text Solution

    |

  3. If the normals to the parabola y^2=4a x at three points (a p^2,2a p), ...

    Text Solution

    |

  4. Normals A O ,A A1a n dA A2 are drawn to the parabola y^2=8x from the p...

    Text Solution

    |

  5. If the normals to the parabola y^2=4a x at the ends of the latus rectu...

    Text Solution

    |

  6. From a point (sintheta,costheta), if three normals can be drawn to the...

    Text Solution

    |

  7. If the normals at P(t(1))andQ(t(2)) on the parabola meet on the same p...

    Text Solution

    |

  8. If the normals to the parabola y^2=4a x at P meets the curve again at ...

    Text Solution

    |

  9. PQ is a normal chord of the parabola y^2 =4ax at P, A being t...

    Text Solution

    |

  10. P ,Q , and R are the feet of the normals drawn to a parabola (y-3)^2=8...

    Text Solution

    |

  11. Normals at two points (x1y1)a n d(x2, y2) of the parabola y^2=4x meet ...

    Text Solution

    |

  12. The endpoints of two normal chords of a parabola are concyclic. Then ...

    Text Solution

    |

  13. If normal at point P on the parabola y^2=4a x ,(a >0), meets it again ...

    Text Solution

    |

  14. The set of points on the axis of the parabola (x-1)^(2)=8(y+2) from wh...

    Text Solution

    |

  15. Tangent and normal are drawn at the point P-=(16 ,16) of the parabola ...

    Text Solution

    |

  16. In parabola y^2=4x, From the point (15,12), three normals are drawn th...

    Text Solution

    |

  17. The line x-y=1 intersects the parabola y^2=4x at A and B . Normals at ...

    Text Solution

    |

  18. If normal are drawn from a point P(h , k) to the parabola y^2=4a x , t...

    Text Solution

    |

  19. The circle x^(2)+y^(2)+2lamdax=0,lamdainR, touches the parabola y^(2)=...

    Text Solution

    |

  20. The radius of the circle whose centre is (-4,0) and which cuts the par...

    Text Solution

    |