The line y = mx + c intersects the parabola la y^2= 4ax at the points P and Q.Show that the coordinates of the mid-point of PQ is ((2a-mc)/(m^2), (2a)/m) If the mid-point is M, find the locus of M whenm varies and c = 1

Question

The line y = mx + c intersects theparabola

y^{2} = 4 a x

at the points P and Q.Show that the coordinates of the mid-point of PQ is

(\frac{2 a - m c}{m^{2}}, \frac{2 a}{m})

.
If the mid-point is M, find the locus of M whenm varies and c = 1

Answer 1

y=mx+c eqn (1)

y^{2} = a x

eqn (2)
Square both side eqn 1
will get

y^{2} = m^{2} x^{2} + 2 m c x + c^{2}

... (3)
set 2 and 3 equals

4 a x = m^{2} x^{2} + 2 m c x + c^{2}

or

(m^{2}) x^{2} + (2 m c - 4 a) x + (c^{2}) = 0

x = - \frac{b}{2 a} = \frac{- (2 m c - 4 a)}{2 m^{2}} = \frac{2 a - m c}{m^{2}}

now plug in function 1

y = m * (\frac{2 a - m c}{m^{2}}) + c = \frac{2 a - m c}{m} + \frac{c m}{m} = \frac{2 a}{m}

so we just showed that it works

(\frac{2 a - m c}{m^{2}}, \frac{2 a}{m})

next,
when c=1:

(\frac{2 a - m}{m^{2}}, \frac{2 a}{m})

Answers (1)