A circle is described whose centre is the vertex and whose diamet… - Vidyadip

MCQ

A circle is described whose centre is the vertex and whose diameter is three-quarters of the latus rectum of the parabola $ y^2 = 4ax.$ If $PQ$ is the common chord of the circle and the parabola and $ L_1 L_2$ is the latus rectum, then the area of the trapezium $PL_1 L_2Q$ is :

A
$3\ \sqrt 2\ a^2$
B
$\left( {\frac{{\sqrt 2 \, + 1}}{2}} \right)\,{a^2}\,$
C
$4 a^2$
✓
$\left( {\frac{{2\,\, + \,\,\sqrt 2 }}{2}} \right) a^2$

✓

Answer

Correct option: D.

$\left( {\frac{{2\,\, + \,\,\sqrt 2 }}{2}} \right) a^2$

d
Centre $(0,0),$ radius $=\frac{1}{2} \cdot \frac{3}{4} \cdot 4 a=\frac{3 a}{2}$

Equation of the circle is $4\left(x^{2}+y^{2}\right)=9 a^{2} \ldots \ldots \ldots$ (i) Equation of the parabola is $y^{2}=4 a x$ $\ldots \ldots$

(ii)

Solving (i) and

(ii) $x^{2}+4 a x-\frac{9 a^{2}}{4}$

$x=\frac{-4 a \pm \sqrt{16 a^{2}+9 a^{2}}}{2}=\frac{-4 a \pm 5 a}{2}$

$\therefore \quad x=a / 2$

$x=a / 2 \quad y^{2}=4 a x=4 a a / 2=2 a^{2}$

$\Rightarrow \quad y=\pm \sqrt{2} a$

$\therefore=2 \sqrt{2} a$

$\therefore \quad=\frac{1}{2}\left(a-\frac{a}{2}\right)(4 a+2 \sqrt{2} \quad a)=\left(\frac{2+\sqrt{2}}{2}\right) a^{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $a_n$ is the greatest term in the sequence $a _{ n }=\frac{ n ^3}{ n ^4+147}, n =1,2,3 \ldots \ldots$. , then $\alpha$ is equal to $..........$.

The equation of the tangent to the parabola ${y^2} = 9x$ which goes through the point $(4, 10)$, is

The mean of $10$ numbers

$7 \times 8,10 \times 10,13 \times 12,16 \times 14, \ldots .$ is ....... .

$\int_{1/e}^e {|\log x|\,dx = } $

An anti aircraft gun take four shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are $0.4, 0.3, 0.2$ and $0.1$ respectively. The probability that the gun hit the plane is :-

Which of the following relations is incorrect

A line drawn through the point $P(4, 7)$ cuts the circle $x^2 + y^2\, = 9$ at the points $A$ and $B$. Then $PA· PB$ is equal to

If $x = 3\,tan\,t$ and $y = 3\,sec\,t,$ then the value of $\frac{{{d^2}y}}{{d{x^2}}}$ at $t = \frac {\pi }{4},$ is

If $f(x) = \frac{2}{{x - 3}},\;g(x) = \frac{{x - 3}}{{x + 4}}$ and $h(x) = - \frac{{2(2x + 1)}}{{{x^2} + x - 12}},$ then $\mathop {\lim }\limits_{x \to 3} [f(x) + g(x) + h(x)]$ is

An infinite number of tangents can be drawn from $(1, 2)$ to the circle ${x^2} + {y^2} - 2x - 4y + \lambda = 0$, then $\lambda = $