Which one of the following curves cuts the parabola $y^{2} = 4 a x$ at right angles

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Solution

The correct option is B

$y^{2} = 4 a x \Rightarrow 2 y (\frac{d y}{d x})_{1} = 4 a \Rightarrow (\frac{d y}{d x})_{1} = \frac{2 a}{y} . . . . (i)$
Taking curve $y = e^{\frac{- x}{2 a}}$
$(\frac{d y}{d x})_{2} = e^{\frac{- x}{2 a}} (- \frac{1}{2 a}) = - \frac{y}{2 a} . . . . (i i)$
Both curves cut orthogonally if,
$(\frac{d y}{d x})_{1} (\frac{d y}{d x})_{2} = - 1 \Rightarrow (- \frac{y}{2 a}) (\frac{2 a}{y}) = - 1$

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Similar questions

Q. Which of the following curves cuts the parabola

y^{2} = 4 a x

at right angle

Q. Draw the curves :

(1) y^{2} = - 4 a x, (2) x^{2} = 4 a y,

and

x^{2} = - 4 a y .

Q. The angle of intersection between the curves

x^{2} = 4 a y

and

y^{2} = 4 a x

at origin is

Match the following equation of parabola

$\begin{matrix} Parabola & parametric equation N) x^{2} = 4 a y & 1) (a t^{2}, 2 a t) E) y^{2} = 4 a x & 2) (- a t, 2 a t) W) y^{2} = - 4 a x & 3) (2 a t, a t^{2}) S) x^{2} = - 4 a y & 4) (2 a t, a t^{2}) \end{matrix}$

Q. Let tangent drawn to the parabola

C_{1} : y^{2} = 4 a x

P

meets the

y

-axis at

Q

. Another parabola

C_{2}

with vertex

Q

and focus

(0, 0)

is drawn which passes through

(2 a, 0) .

Identify which of the following statements can be CORRECT?

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Which one of the following curves cuts the parabola y2=4ax at right angles

The correct option is B y2=4ax⇒2y(dydx)1=4a⇒(dydx)1=2ay ....(i) Taking curve y=e−x2a (dydx)2=e−x2a(−12a)=−y2a ....(ii) Both curves cut orthogonally if, (dydx)1(dydx)2=−1⇒(−y2a)(2ay)=−1

Which one of the following curves cuts the parabola $y^{2} = 4 a x$ at right angles