Question:
The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:
The line \( x - 2y - 3 = 0 \) cuts the parabola \( y^2 = 4ax \) at points P and Q. If the focus of this parabola is \( \left(\frac{1}{4}, k\right) \), then PQ is:
Show Hint
For chord length problems in parabolas, use intersection substitution and standard chord length formulas.
Updated On: May 19, 2025
- \( 16a\sqrt{5} \)
- \( 8a\sqrt{5} \)
- \( 4a\sqrt{5} \)
- \( 2a\sqrt{5} \)
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Step 1: Finding the intersection points
We substitute \( x = \frac{y + 3}{2} \) into the parabola equation \( y^2 = 4ax \):
\[
y^2 = 4a \left( \frac{y + 3}{2} \right)
\]
\[
y^2 - 2ay - 6a = 0
\]
Solving this quadratic equation in \( y \) gives the points \( P(y_1) \) and \( Q(y_2) \).
Step 2: Finding the distance PQ
Using the chord length formula:
\[
PQ = \frac{|2a|}{\sqrt{1 + (m^2)}}
\]
where \( m = 2 \) (slope of line),
\[
PQ = \frac{2a \times \sqrt{5}}{1}
\]
\[
PQ = 16a\sqrt{5}
\]
Was this answer helpful?
2
0
Top Questions on Parabola
- Let A and B be the two points of intersection of the line \( y + 5 = 0 \) and the mirror image of the parabola \( y^2 = 4x \) with respect to the line \( x + y + 4 = 0 \). If \( d \) denotes the distance between A and B, and \( a \) denotes the area of \( \Delta SAB \), where \( S \) is the focus of the parabola \( y^2 = 4x \), then the value of \( (a + d) \) is:
Two parabolas have the same focus $(4, 3)$ and their directrices are the $x$-axis and the $y$-axis, respectively. If these parabolas intersect at the points $A$ and $B$, then $(AB)^2$ is equal to:
- 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:
- If the line \( 3x - 2y + 12 = 0 \) intersects the parabola \( 4y = 3x^2 \) at the points A and B, then at the vertex of the parabola, the line segment AB subtends an angle equal to:
- Let the point $ P $ of the focal chord $ PQ $ of the parabola $ y^2 = 16x $ be $ (1, -4) $. If the focus of the parabola divides the chord $ PQ $ in the ratio $ m : n $, gcd($m, n$) = 1, then $ m^2 + n^2 $ is equal to:
View More Questions
Questions Asked in AP EAMCET exam
- The angle made by the resultant vector of two vectors \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( 2\hat{i} - 7\hat{j} - 4\hat{k} \) with the x-axis is:
- If the longest wavelength of the spectral line of the Paschen series of \( \text{Li}^{2+} \) ion spectrum is \( x \) Å, then the longest wavelength (in Å) of the Lyman series of the hydrogen spectrum is:
- AP EAMCET - 2024
- Atomic Spectra
- Let \( A(2, 3), B(1, -1) \) be two points. If \( P \) is a variable point such that \( \angle APB = 90^\circ \), then the locus of \( P \) is
- AP EAMCET - 2024
- solution of system of linear inequalities in two variables
- The charge \( q \) (in coulombs) passing through a \( 10 \Omega \) resistor as a function of time \( t \) (in seconds) is given by: \[ q = 3t^2 - 2t + 6 \] The potential difference across the ends of the resistor at time \( t = 5 \) s is:
- AP EAMCET - 2024
- coulombs law
The percentage error in the measurement of mass and velocity are 3% and 4% respectively. The percentage error in the measurement of kinetic energy is:
- AP EAMCET - 2024
- Projectile motion
View More Questions