Question

If normal are drawn from a point `P(h , k)` to the parabola `y^2=4a x` , then the sum of the intercepts which the normals cut-off from the axis of the parabola is `(h+c)` (b) `3(h+a)` `2(h+a)` (d) none of these

Answer 1

let `x_1, x_2, x_3` be the intercept on the axis of parabola
eqn of normal `y = mx-2am-am^3`
`y=0`
`mx= 2am+ am^3`
`x= 2a+am^2`
`x_1=2a+ am_1^2`
`x_2= 2a+am_2^2 & x_3= 2a+ am_3^2`
eqn of normal at p(h,k)
`y= mx-2am-am^3`
`k=hm-2am-am^3`
`am^3 +(2a-h)m+k=0 `
`m_1+m_2+m_3= -b/a = 0`
`m_1m_2 + m_2m_3 + m_3m_1 = c/a = (2a-h)/a
`m_1^2 + m_2^2 + m_3^2 = (m_1+m_2+m_3)^2 - 2(m_1m_2 + m_2m_3 + m_3m_1)
`(0)^2 - 2((2a-h)/a)`
`-2((2a-h)/a)`
`x_1 + x_2 + x_3 = 6a + a(m_1^2 + m_2^2 +m_3^2)`
`= 6a + cancel(a)(-2((2a-h))/cancel(a))`
`=6a-4a+2h = 2a+2h = 2(a+h)` answer