Question

If the chord of contact of tangents from a point p to the parabola y2=4ax touches the parabola x2=4 by, then find the locus of p.

Answer 1

The locus x y = - 2 a b , i.e Hyperbola

Step-by-step explanation:

Given as :

The equation of parabola one , y² = 4 a x             ...........1

The equation of parabola two, x² = 4 b y               .........2

The chord of contact of tangents from a point p to the parabola one touches the parabola two

Let The point of contact at p = h , k

According to question

The chord is tangent to the parabola 2

The equation of chord of contact =  k y = 2 a ( x + h )

Or,                                                        y = $\dfrac{2a}{k}$ ( x + h )

From equation 2

x² =  4 b ( $\dfrac{2a}{k}$) ( x + h ) )

Or, x²  $(\dfrac{8ab}{k} )$² - 4 x (1) ( $\dfrac{-8abh}{k}$ ) = 0

Or,  [ $\dfrac{64a^{2}b^{2} }{h}$ + h ] [ $\dfrac{2ab}{k}$ + h ] = 0

Or, [ $\dfrac{2ab}{k}$ + h ] = 0

Or, $\dfrac{2ab}{k}$ = - h

or, 2 a b = - h k

∴ locus pf point p at ( h , k ) is x y = - 2 a b

So, The locus at point p is Hyperbola

Hence, The locus x y = - 2 a b , i.e Hyperbola   Answer

Answer 2

Answer:

The locus x y = - 2 a b , i.e Hyperbola

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