Maharashtra State BoardHSC Science (Electronics) 11th
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Show that the line 8y + x = 17 touches the ellipse x2 + 4y2 = 17. Find the point of contact - Mathematics and Statistics

Sum

Show that the line 8y + x = 17 touches the ellipse x2 + 4y2 = 17. Find the point of contact

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Solution

Given equation of the ellipse is x2 + 4y2 = 17.

∴ `x^2/17 + y^2/(17/4)` = 1

Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1 we get

a2 = 17 and b2 = `17/4`

Given equation of line is 8y + x = 17,

i.e., y = `(-1)/8 "x" + 17/8`

Comparing this equation with y = mx + c, we get

m = `(-1)/8` and c = `17/8`

For the line y = mx + c to be a tangent to the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1, we must have

c2 = a2 m2 + b2

c2 = `(17/8)^7 = 289/64`

a2m2 + b2 = `17((-1)/8)^2 + 17/4`

= `17/64 + 17/4`

= `289/64`

= c2

∴ The given line touches the given ellipse and point of contact is

`((-"a"^2"m")/"c", "b"^2/"c") = ((-17((-1)/8))/(17/8), (17/4)/(17/8))`

= (1, 2).

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APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 7 Conic Sections
Miscellaneous Exercise 7 | Q 2.19 | Page 178
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