Question

9. If (a,a-1) lies inside the ellipse
16x? +9y2 - 16x =0 then a lies in the
9
interval
k
. Find k.
25
1)1
2) 2
3)3
4)4​

Answer 1

Concept-

To check if points lies inside the ellipse, put the point values in the equation of ellipse, if the equation satisfies then it lies on the plane.

Given-

The equation of ellipse is 16x²+9y²-16x and ∝ lies in the interval (9/25, k).

Find-

What is the value of k?

Solution-

f(∝,∝-1)<0

16∝²+9(∝²-2∝+1)-16∝+9<0

16∝²+9∝²-18∝-16∝+9<0

25∝²-34∝+9<0

25∝²-25∝-9∝+9

25∝(∝-1)-9(∝-1)<0

(∝-1)(25∝-9)<0

put ∝-1=0 and 25∝-9=0

⇒ ∝=1 and ∝=9/25

⇒∝∈(9/25,1)

So, the value of k is 1, i.e. K=1

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