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Question
Find the equation of the tangent to the ellipse `x^2/25 + y^2/4` = 1 which are parallel to the line x + y + 1 = 0.
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Solution
Given equation of the ellipse is `x^2/25 + y^2/4` = 1
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 25 and b2 = 4
Slope of the given line x + y + 1 = 0 is –1.
Since the given line is parallel to the required tangents, slope of the required tangents is m = –1.
Equations of tangents to the ellipse
`x^2/"a"^2 + y^2/"b"^2` = 1 having slope m are
y = `"m"x ± sqrt("a"^2"m"^2 + "b"^2)`
∴ y = `-x ± sqrt(25(-1)^2 + 4)`
∴ y = `-x ± sqrt(29)`
∴ x + y = `± sqrt(29)`.
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