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in Mathematics by (67.9k points)

The points, where the normal to the ellipse x+ 3y2 = 37 be parallel to the line 6x – 5y + 7 = 0 is / are

(a)  (5,2)

(b)  (2,5)

(c)  (1,3)

(d)  (–5,–2)

1 Answer

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Best answer

Correct option  (a) and (b)

Explanation:

Let (x1 ,y1 ) be a point on ellipse then, x1+ 3y1 2 = 37  ...(1)

Equation of normal at (x1 ,y1 ) is a2x/x1 - b2y/y1 = a2 - b2

In ellipse x2/37 + y2/37/3 = 1

a2 = 37, b2 = 37/3

37x/x1  - 37y/3y1 = 37 - 37/3

37x/x1  - 37y/3y1 = 74/3  ...(2)

Slope of normal is

it is parallel to 6x – 5y + 7 = 0

3y1/x1 = 6/5

y1 = 2/5 x1

put in (1) we get

x21 + 12/25 x21 = 37

37x21/25 = 37

x21 = 25

x1  = ± 5

y1 =± 2

(5,2) & (–5,–2)

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