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Question
The line lx + my + n = 0 will touch the parabola y2 = 4ax if ln = am2.
Options
True
False
Solution
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`
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