the equation of the common tangent to the equal parabolas y

The equation of the common tangent to the equal parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$ is :

A. $$x + y + a = 0$$

B. $$x + y = a$$

C. $$x - y = a$$

D. none of these

Answer : $$x + y + a = 0$$

Solution :
Any tangent to $${y^2} = 4ax$$ is $$y = mx + \frac{a}{m}.$$ It touches $${x^2} = 4ay$$ if $${x^2} = 4a\left( {mx + \frac{a}{m}} \right),$$ i.e., $${x^2} - 4amx - \frac{{4{a^2}}}{m} = 0$$ has equal roots.
So, $${m^3} + 1 = 0,$$ i.e., $$m = - 1.$$ Hence, the common tangent is $$y = - x - a.$$

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

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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

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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

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

View Answer

The equation of the common tangent to the equal parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$ is :

Releted MCQ Question on
Geometry >> Parabola

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-

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

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

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-

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Parabola

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The equation of the common tangent to the equal parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$ is :

Releted MCQ Question on Geometry >> Parabola

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-

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

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

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-

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Geometry >> Parabola

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