# Show that the line x – y = 5 is a tangent to the ellipse 9x2 + 16y2 = 144. Find the point of contact - Mathematics and Statistics

Sum

Show that the line x – y = 5 is a tangent to the ellipse 9x2 + 16y2 = 144. Find the point of contact

#### Solution

Given equation of the ellipse is 9x2 + 16y2 = 144

∴ x^2/16 + y^2/9 = 1

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

a2 = 16 and b2 = 9

Given equation of line is x – y = 5, i.e., y = x – 5

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

m = 1 and c = – 5

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 = a2m2 + b2

c2 = (–5)2 = 25

a2m2 + b2 = 16(1)2 + 9 = 16 + 9 = 25 = c2

∴ The given line is a tangent to the given ellipse and point of contact

= ((-"a"^2"m")/"c", "b"^2/"c")

= (((-16)(1))/-5, 9/-5)

= (16/5, (-9)/5).

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