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1

### AIEEE 2006

Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is
A
$$\pi$$
B
$${\pi \over 2}$$
C
$${\pi \over 6}$$
D
$${\pi \over 4}$$

## Explanation

$${{dy} \over {dx}} = 2x - 5$$

$$\therefore$$ $${m_1} = {\left( {2x - 5} \right)_{\left( {2,0} \right)}} = - 1,$$

$${m_2} = {\left( {2x - 5} \right)_{\left( {3,0} \right)}} = 1 \Rightarrow {m_1}{m_2} = - 1$$

i.e. the tangents are perpendicular to each other.
2

### AIEEE 2006

In the ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is
A
$${3 \over 5}$$
B
$${1 \over 2}$$
C
$${4 \over 5}$$
D
$${1 \over {\sqrt 5 }}$$

## Explanation

$$2ae = 6 \Rightarrow ae = 3;\,\,2b = 8 \Rightarrow b = 4$$

$${b^2} = {a^2}\left( {1 - {e^2}} \right);16 = {a^2} - {a^2}{e^2}$$

$$\Rightarrow a{}^2 = 16 + 9 = 25$$

$$\Rightarrow a = 5$$

$$\therefore$$ $$e = {3 \over a} = {3 \over 5}$$
3

### AIEEE 2006

The locus of the vertices of the family of parabolas
$$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is
A
$$xy = {{105} \over {64}}$$
B
$$xy = {{3} \over {4}}$$
C
$$xy = {{35} \over {16}}$$
D
$$xy = {{64} \over {105}}$$

## Explanation

Given parabola is $$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$

$$\Rightarrow y = {{{a^3}} \over 3}\left( {{x^3} + {3 \over {2a}} + x + {9 \over {16{a^2}}}} \right) - {{3a} \over {16}} - 2a$$

$$\Rightarrow y + {{35a} \over {16}} = {{{a^3}} \over 3}{\left( {x + {3 \over {4a}}} \right)^2}$$

$$\therefore$$ Vertex of parabola is $$\left( {{{ - 3} \over {4a}},{{ - 35a} \over {16}}} \right)$$

To find locus of this vertex,

$$x = {{ - 3} \over {4a}}\,\,$$ and $$\,\,y = {{ - 35a} \over {16}}$$

$$\Rightarrow a = {{ - 3} \over {4x}}\,\,$$ and $$a = - {{16y} \over {35}}$$

$$\Rightarrow {{ - 3} \over {4x}} = {{ - 16y} \over {35}} \Rightarrow 64xy = 105$$

$$\Rightarrow xy = {{105} \over {64}}$$ which is the required locus.
4

### AIEEE 2005

The locus of a point $$P\left( {\alpha ,\beta } \right)$$ moving under the condition that the line $$y = \alpha x + \beta$$ is tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is
A
an ellipse
B
a circle
C
a parabola
D
a hyperbola

## Explanation

Tangent to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ is

$$y = mx \pm \sqrt {{a^2}{m^2} - {b^2}}$$

Given that $$y = \alpha x + \beta$$ is the tangent of hyperbola

$$\Rightarrow m = \alpha$$ and $${a^2}{m^2} - {b^2} = {\beta ^2}$$

$$\therefore$$ $${a^2}{\alpha ^2} - {b^2} = {\beta ^2}$$

Locus is $${a^2}{x^2} - {y^2} = {b^2}$$ which is hyperbola.

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