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At a point P on the parabola x^2=4ay, a normal PK is drawn. From the vertex O a perpendicular OM is drawn to meet the normal at M. Show that the equation of the locus of M as P varies on the parabola is x^4 2ax^2y + x^2y^2  ay^3 = 0
 2 years ago
 2 years ago
At a point P on the parabola x^2=4ay, a normal PK is drawn. From the vertex O a perpendicular OM is drawn to meet the normal at M. Show that the equation of the locus of M as P varies on the parabola is x^4 2ax^2y + x^2y^2  ay^3 = 0
 2 years ago
 2 years ago

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Ishaan94Best ResponseYou've already chosen the best response.2
\[x^2 = 4ay\] dw:1319018817978:dw Differentiating with respect to x \[2x = 4a \frac{dy}{dx}\] \[\frac{x}{2a} =\frac{dy}{dx}\] Normal is perpendicular to tangent Hence, Slope of Normal is \(\frac{2a}{x}\) @mimi can you draw the figure? I am confused
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.3
lols, im confused as well, i've attempted on drawing it but i think that it's wrong =/
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
'From the vertex O a perpendicular OM is drawn to meet the normal at M' Maybe something is not right here, I am not sure but try reading the question again maybe some typo
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
ah Okay I think I got it
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
maybe the figure is like thisdw:1319019992221:dw
 2 years ago

Kira_YamatoBest ResponseYou've already chosen the best response.0
dw:1319020480071:dw
 2 years ago

Kira_YamatoBest ResponseYou've already chosen the best response.0
dw:1319020633975:dw
 2 years ago

Kira_YamatoBest ResponseYou've already chosen the best response.0
dw:1319020681433:dw
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
equation of normal must be according to the y = mx + c form \[y' = \frac{2a}{x}x' + c \]Now we have points (x,y) for the Normal on parabola \(x^2 = 4ay\) \[y = \frac{2a}{x}x+c\] \[\frac{x^2}{4a} = 2a + c \] \[ \frac{x^2}{8a^2} = c\]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
sorry, typo ignore the last expression
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
\[\frac{x^2 + 2a*4a}{4a}=c\] \[\frac{x^2 + 8a^2}{4a}=c\]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
OM is from origin perpendicular to PK equation of OM \[y' = \frac{x}{2a}x'\]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
equation of PK \[y' = x' \frac{2a}{x} + \frac{x^2 + 8a^2}{4a} \]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
The point M satisfies both the equation i.e equation of PK and equation of OM
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
Let M be (x_1,y_1) \[y_1 = x_1* \frac{2a}{x} + \frac{x^2 + 8a^2}{4a}\] \[y_! = x_1 *\frac{x}{2a}\] \[ {x_1*}\frac{2a}{x} + \frac{x^2 + 8a^2}{4a} = \frac{x}{2a}*x_1\] \[ x_1 \left( \frac{4a^
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
\[x_1 \left( \frac{4a^2 + x^2 }{2ax}\right) = \frac{x^2 + 8a^2}{4a}\]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
\[x_1 = \frac{x^3 + 8xa^2}{8a^2 + 2x^2}\]
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
\[y_1 = \frac{x}{2a}\times \frac{x^3 + 8a^2x}{8a^2 +2x^2} \]
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.3
I'm lost =/ Isn't the equation of the normal y = x + 2ap^3/p
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
I don't think so \[ x^2 = 4ay \] \[\frac{dy}{dx} = \frac{x}{2a}\] Slope of tangent, Normal is perpendicular to tangent
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.2
This is getting complicated, I will try this later I think I am getting the question wrong. I will try this on my notebook first
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.3
It's okay never mind then, I will ask my teacher tomorrow.
 2 years ago
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