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Let y^(2) -5y +3x +k = 0 be a parabola, ...

Let `y^(2) -5y +3x +k = 0` be a parabola, then

A

its latus rectum is least when `k = 1`

B

its latus rectum is independent of k

C

the line `y = 2x +1` will touch the parabola if `k = (73)/(16)`

D

`y = (5)/(2)` is the only normal to the parabola whose slope is zero

Text Solution

Verified by Experts

The correct Answer is:
B, C, D
The equation to the parabola can be written as:
`(y-(5)/(2))^(2) =- 3 (x-(25-4k)/(12))`
The length of the latus rectum is `3.y = 2x+1` is a tangent. So, the quadratic equation `(2x+1)^(2) - 5 (2x+1) +3x +k =0` or `4x^(2) -3x +k-4 =0` must have equal roots. Now roots are equal if `b^(2) = 4ac`
`rArr 9 = 16(k-4) rArr k =(73)/(16)`
`y = (5)/(2)` is normal at the vertex which has slope 0.
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