Question

If 2x + 3y= c, is normal to y^2 = 9x, then value of c is?

Answer 1

Answer: The answer is 11.

Step-by-step explanation:  Given equations of the parabola and the line are

$2x+3y=c\\\\\Rightarrow y=-\dfrac{2}{3}x+\dfrac{c}{3}\\\\\Rightarrow y=mx+C,$

and

$y^2=9x\\\\\Rightarrow y^2=4\times \dfrac{9}{4}x\\\\\Rightarrow y^2=4ax.$

Here,

$a=\dfrac{9}{4},~~m=-\dfrac{2}{3},~~C=\dfrac{c}{3}.$

Now, we know that the line y = mx + C will be a normal to the parabola y² = 4ax if

$C=-2am-am^3\\\\\Rightarrow \dfrac{c}{3}=-2\times\dfrac{9}{4}\times (-\dfrac{2}{3})-\dfrac{9}{4}\times(-\dfrac{2}{3})^3\\\\\\\Rightarrow \dfrac{c}{3}=3+\dfrac{2}{3}\\\\\Rightarrowc=11.$

Thus, the answer is c = 11.