if the line x 10 is the directrix of the parabola y 2 kx 8

If the line $$x-1=0$$ is the directrix of the parabola $${y^2} - kx + 8 = 0,$$ then one of the values of $$k$$ is-

A. $$\frac{1}{8}$$

B. $$8$$

C. $$4$$

D. $$\frac{1}{4}$$

Answer : $$4$$

Solution :
KEY CONCEPT : The directrix of the parabola $${y^2} = 4a$$
$$\left( {x - {x_1}} \right)$$ is given by $$x = {x_1} - a$$
$${y^2} = kx - 8 \Rightarrow {y^2} = k\left( {x - \frac{8}{k}} \right)$$
Directrix of parabola is $$x = \frac{8}{k} - \frac{k}{4};$$
Now, $$x=1$$ also coincides with $$x = \frac{8}{k} - \frac{k}{4}$$
On comparison, $$\frac{8}{k} - \frac{k}{4} = 1,$$ or $${k^2} - 4k - 32 = 0$$
On solving we get $$k=4$$

Releted MCQ Question on
Geometry >> Parabola

Releted Question 1

Consider a circle with its centre lying on the focus of the parabola $${y^2} = 2px$$ such that it touches the directrix of the parabola. Then a point of intersection of the circle and parabola is-

A. $$\left( {\frac{p}{2},\,p} \right){\text{ or }}\left( {\frac{p}{2},\, - p} \right)$$

B. $$\left( {\frac{p}{2},\, - \frac{p}{2}} \right)$$

C. $$\left( { - \frac{p}{2},\,p} \right)$$

D. $$\left( { - \frac{p}{2},\, - \frac{p}{2}} \right)$$

View Answer

Releted Question 2

The curve described parametrically by $$x = {t^2} + t + 1,\,\,y = {t^2} - t + 1$$ represents-

A. a pair of straight lines

B. an ellipse

C. a parabola

D. a hyperbola

View Answer

Releted Question 3

If $$x+y=k$$ is normal to $${y^2} = 12x,$$ then $$k$$ is-

A. $$3$$

B. $$9$$

C. $$ - 9$$

D. $$ - 3$$

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Releted Question 4