# 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

#### 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