the locus of the mid point of the line segment joining the

The locus of the mid-point of the line segment joining the focus to a moving point on the parabola $${y^2} = 4ax$$ is another parabola with directrix :

A. $$x = - a$$

B. $$x = - \frac{a}{2}$$

C. $$x = 0$$

D. $$x = \frac{a}{2}$$

Answer : $$x = 0$$

Solution :
If $$\left( {h,\,k} \right)$$ is the mid point of line joining focus $$\left( {a,\,0} \right)$$ and $$Q\left( {a{t^2},\,2at} \right)$$ on parabola then $$h = \frac{{a + a{t^2}}}{2},\,k = at$$
Eliminating $$t,$$ we get $$2h = a + a\left( {\frac{{{k^2}}}{{{a^2}}}} \right)$$
$$ \Rightarrow {k^2} = a\left( {2h - a} \right)\,\,\,\, \Rightarrow {k^2} = 2a\left( {h - \frac{a}{2}} \right)$$
$$\therefore $$ Locus of $$\left( {h,\,k} \right)$$ is $${y^2} = 2a\left( {x - \frac{a}{2}} \right)$$
whose directrix is $$\left( {x - \frac{a}{2}} \right) = - \frac{a}{2}$$
$$ \Rightarrow x = 0$$

Releted MCQ Question on
Geometry >> Locus

Releted Question 1

The equation $$\frac{{{x^2}}}{{1 - r}} - \frac{{{y^2}}}{{1 + r}} = 1,\,\,\,r > 1$$ represents :

A. an ellipse

B. a hyperbola

C. a circle

D. none of these

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Releted Question 2

The equation $$2{x^2} + 3{y^2} - 8x - 18y + 35 = k$$ represents :

A. no locus if $$k>0$$

B. an ellipse if $$k<0$$

C. a point if $$k=0$$

D. a hyperbola if $$k>0$$

View Answer

Releted Question 3

If $$a>2b>0$$ then the positive value of $$m$$ for which $$y = mx - b\sqrt {1 + {m^2}} $$ is a common tangent to $${x^2} + {y^2} = {b^2}$$ and $${\left( {x - a} \right)^2} + {y^2} = {b^2}$$ is :

A. $$\frac{{2b}}{{\sqrt {{a^2} - 4{b^2}} }}$$

B. $$\frac{{\sqrt {{a^2} - 4{b^2}} }}{{2b}}$$

C. $$\frac{{2b}}{{a - 2b}}$$

D. $$\frac{b}{{a - 2b}}$$

View Answer

Releted Question 4