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Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola x^2/7-y^2/5=1 such that the sum of the cubes of their slopes is 8. - Mathematics and Statistics

Find the equation of the locus of the point of intersection of two tangents drawn to the hyperbola `x^2/7-y^2/5=1` such that the sum of the cubes of their slopes is 8. 

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Solution

Let ` p(x_1,y_1) ` be a point on the locus. The equation of the hyperbola is `x^2/7-y^2/5=1` for which `a^2=7, b^2=5`

The equation of the tangents to the hyperbola `x^2/a^2-y^2/b^2 =1` in terms of slope m are

`y=mx+-sqrt(a^2m^2-b^2)`

the equation of the tangents to the given hyperbola are

`y=mx+-sqrt(7m^2-5)`

If these tangents pass through the point `p(x_1,y_1) ` we get 

`y_1=mx_1+-sqrt(7m^2-5)`

y

`y_1-mx_1=+-sqrt(7m^2-5)`

On squaring both sides, we get,

`(y_1-mx_1)^2=7m^2-5`

`y_1^2-2mx_1y_1+m^2x_1^2=7m^2-5`

`(x_1^2-7)m^2-2mx_1y_1+(y_1^2+5)=0`

The roots `m_1,m_2` of this quadratic equation in m are the slopes of the tangents drawn from the point P to the
hyperbola.
From this quadratic equation,

`{ (m_1+m_2=(2x_1y_1)/(x_1^2-7)),(and m_1m_2=(y_1^2+5)/(x_1^2-7)) :} ...........(1)`

We are given that,
sum of the cubes of the slopes = 8

`m_1^3+m_2^3=8`

`(m_1+m_2)^3-3m_1m_2(m_1+m_2)=8`

`((2x_1y_1)/(x_1^2-7))^3-3((y_1^2+5)/(x_1^2-7))((2x_1y_1)/(x_1^2-7))=8 .................by(1)`

`8x_1^3y_1^3-6x_1y_1(y_1^2+5)(x_1^2-7)=8(x_1^2-7)^3`

`8x_1^3y_1^3-6x_1y_1(y_1^2+5)(x_1^2-7)-8(x_1^2-7)^3=0`

the equation of the locus of ` P(x_1,y_1)` is

`8x^3y^3-6xy(y_1^2+5)(x_1^2-7)-8(x_1^2-7)^3=0`

Concept: Conics - Locus of Points from Which Two Tangents Are Mutually Perpendicular
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