Let \( P(4, 4\sqrt{3}) \) be a point on the parabola \( y^2 = 4ax \) and PQ be a focal chord of the parabola. If M and N are the foot of the perpendiculars drawn from P and Q respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to:
Show Hint
- \( \frac{263\sqrt{3}}{8} \)
- \( \frac{343\sqrt{3}}{8} \)
- \( \frac{34\sqrt{3}}{3} \)
- \( 17\sqrt{3} \)
The Correct Option is A
Solution and Explanation
Given that \( P(4, 4\sqrt{3}) \) lies on the parabola \( y^2 = 4ax \), we can find \( a \) by substituting the coordinates of P: \[ (4\sqrt{3})^2 = 4a(4) \quad \Rightarrow \quad 48 = 16a \quad \Rightarrow \quad a = 3. \] Now, since PQ is a focal chord, we use the standard result for the area of a quadrilateral formed by the focal chord and perpendiculars from the points on the parabola to the directrix.
The area of quadrilateral PQMN is \( \frac{263\sqrt{3}}{8} \).
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Parabola Questions
- \((α, β)\) lie on the parabola \(y^2 = 4x\) and \((α, β)\) also lie on chord with midpoint \((1,\frac 54)\) of another parabola \(x^2 = 8y\), then value of \(|(8 – β)(α – 28)|\) is
- The area of the region enclosed by the parabolas \(y = 4 – x^2\) and \(3y = (x – 4)^2\) is in (sq. unit)?
- The area of the region enclosed by the parabola \((y - 2)^2 = x - 1\), the line \(x - 2y + 4 = 0\) and the positive coordinate axes is ______.
If the shortest distance of the parabola \(y^{2}=4x\) from the centre of the circle \(x² + y² - 4x - 16y + 64 = 0\) is d, then d2 is equal to:
- Let the length of the focal chord \( PQ \) of the parabola \( y^2 = 12x \) be 15 units. If the distance of \( PQ \) from the origin is \( p \), then \( 10p^2 \) is equal to _____
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.