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Question
Tangents are drawn through a point P to the ellipse 4x2 + 5y2 = 20 having inclinations θ1 and θ2 such that tan θ1 + tan θ2 = 2. Find the equation of the locus of P.
Solution
Given equation of the ellipse is 4x2 + 5y2 = 20.
∴ `x^2/5 + y^2/4` = 1
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 5 and b2 = 4
Since inclinations of tangents are θ1 and θ2 ,
m1 = tan θ1 and m2 = tan θ2.
Equation 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 = `"m"x ± sqrt(5"m"^2 + 4)`
∴ y – mx = `± sqrt(5"m"^2 + 4)`
Squaring both the sides, we get
y2 – 2mxy + m2x2 = 5m2 + 4
∴ (x2 – 5)m2 – 2xym + (y2 – 4) = 0
The roots m1 and m2 of this quadratic equation are the slopes of the tangents.
∴ m1 + m2 = `(-(-2xy))/(x^2 - 5) = (2xy)/(x^2 - 5)`
Given, tan θ1 + tan θ2 = 2
∴ m1 + m2 = 2
∴ `(2xy)/(x^2 - 5)` = 2
∴ xy = x2 – 5
∴ x2 – xy – 5 = 0, which is the required equation of the locus of P.
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