Question

The line lx + my + n = 0 will touch the parabola y² = 4ax if ln = am².

पर्याय

• True

• False

Answer 1

This statement is True.

Explanation:

Given equation of parabola is y² = 4ax  .....(i)

And the equation of line is lx + my + n = 0  ......(ii)

From equation (ii), we have

y = `(-lx - n)/m`

Putting the value of y in equation (i) we get

`((-lx - n)/m)^2 = 4ax`

⇒ `l^2x^2 + n^2 + 2lnx - 4am^2x` = 0

⇒ `l^2x^2 + (2ln - 4am^2)x + n^2` = 0

If the line is the tangent to the circle

Then `b^2 - 4ac` = 0

`(2ln - 4am^2) - 4l^2n^2` = 0

⇒ `4l^2n^2 + 16a^2m^4 - 16lnm^2a - 4l^2n^2` = 0

⇒ `16a^2m^4 - 16lnm^2a` = 0

⇒ `16am^2(am^2 - ln)` = 0

⇒ `am^2(am^2 - ln)` = 0

⇒ `am^2 ≠ 0`

∴ `am^2 - ln` = 0

∴ `ln - am^2`