p is a point two tangents are drawn from it to the parabola

$$P$$ is a point. Two tangents are drawn from it to the parabola $${y^2} = 4x$$ such that the slope of one tangent is three times the slope of the other. The locus of $$P$$ is :

A. a straight line

B. a circle

C. a parabola

D. an ellipse

Answer : a parabola

Solution :
Let $$P = \left( {\alpha ,\,\beta } \right).$$ Any tangent to the parabola is $$y = mx + \frac{a}{m}.$$ It passes through $$\left( {\alpha ,\,\beta } \right).$$ So, $$\beta = m\alpha + \frac{1}{m}\left( {\because {\text{ here }}a = 1} \right);\,\,\therefore \,{m^2}\alpha - \beta m + 1 = 0$$
Its roots are $${m_1},\,3{m_1}.$$ So, $${m_1} + 3{m_1} = \frac{\beta }{\alpha },\,\,\,{m_1}.3{m_1} = \frac{1}{\alpha }$$
$$\therefore \,\,3.{\left( {\frac{\beta }{{4\alpha }}} \right)^2} = \frac{1}{\alpha }{\text{ or }}3{\beta ^2} = 16\alpha $$
Thus, the locus is $$3{y^2} = 16x,$$ which is a parabola.

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. $$ - 3$$

View Answer

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

$$P$$ is a point. Two tangents are drawn from it to the parabola $${y^2} = 4x$$ such that the slope of one tangent is three times the slope of the other. The locus of $$P$$ 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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$$P$$ is a point. Two tangents are drawn from it to the parabola $${y^2} = 4x$$ such that the slope of one tangent is three times the slope of the other. The locus of $$P$$ 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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