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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Parabola

NTA Abhyas 2020 · Mathematics · Parabola

If the line y = - (7)/(2) is the directrix of the parabola x ^ 2 -...

If the line y=−72y = - \dfrac{7}{2} is the directrix of the parabola x2−ky+8=0{x}^{2} - ky + 8 = 0, then the sum of all the possible values of kk is equal to

Correct answer

14

Step-by-step solution

Given equation of the parabola is x2−ky+8=0⇒x2=k(y−8k){x}^{2} - ky + 8 = 0 ⇒ {x}^{2} = k\left(y - \dfrac{8}{k}\right)

⇒⇒ Directrix is y=8k−k4y = \dfrac{8}{k} - \dfrac{k}{4}

⇒y=−72⇒ y = - \dfrac{7}{2} is same as y=8k−k4y = \dfrac{8}{k} - \dfrac{k}{4}

⇒32−k24k=−72⇒−k2+14k+32=0⇒ \dfrac{32 - {k}^{2}}{4k} = - \dfrac{7}{2} ⇒ - {k}^{2} + 14k + 32 = 0

⇒k1+k2=14⇒ {k}_{1} + {k}_{2} = 14 (sum of possible values of kk)
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