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in the given figure y=x^(2)+bx+c is a qu...


in the given figure `y=x^(2)+bx+c` is a quadratic polynomial which meets x-axis at A and B and y-axis at C Q is vertex of quadratic polynomial and foot of perpendicular from Q to x-axis and y-axis are P and R respectively. Then answer the following questions.
Q. If `("area of" DeltaABC)/("Area of rectangle" OPQR)=(8)/(3)` satisfies the relation `b^(2)=kc` then `k` will be

A

`9`

B

`(2)/(9)`

C

`(9)/(2)`

D

`2`

Text Solution

Verified by Experts

The correct Answer is:
B
`((1)/(2)(beta-alpha)c)/(((b^(2)-4c)/(4))(-(b)/(2)))=(8)/(3)` squaring and using
`(beta-alpha)^(2)=b^(2)-4c`
`9c^(2)=4b^(4)-16b^(2)c`
`4b^(4)-16b^(2)c-9c^(2)=0`
`4b^(4)-18b^(2)c+2b^(2)c-9c^(2)=0`
`(2b^(2)+c)(2b^(2)-9c)=0impliescgt0because2b^(2)+cne0`
`(b^(2))/(c)=(9)/(2)implies(b^(2))/(c)=4.5`
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Knowledge Check

  • in the given figure y=x^(2)+bx+c is a quadratic polynomial which meets x-axis at A and B and y-axis at C Q is vertex of quadratic polynomial and foot of perpendicular from Q to x-axis and y-axis are P and R respectively. Then answer the following questions. Q. if two circles passing through A and B touches y-axis at C and R respectively then

    A
    `b^(2)=2c`
    B
    `b^(2)=3c`
    C
    `b^(2)=4c`
    D
    `b^(2)=8c`

  • A quadratic polynomial maps from [-2, 3] onto [0, 3] and touches x-axis at x = 3, then the polynomial is

    A
    `(3)/(16)(x^(2)-6x+16)`
    B
    `(3)/(25)(x^(2)-6x+9)`
    C
    `(3)/(25)(x^(2)-6x+16)`
    D
    `(3)/(16)(x^(2)-6x+9)`

  • From the point (4, 3) a perpendicular is dropped on the x-axis well as on the y-axis. If the lengths of perpendiculars are p, q respectively, then which one of the following is correct ?

    A
    `p=q`
    B
    `3p=4q`
    C
    `4p=3q`
    D
    `p+q=5`

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