Advertisements
Advertisements
Question
In the parabola y2 = 4ax, the length of the chord passing through the vertex and inclined to the axis at π/4 is
Options
\[4\sqrt{2}a\]
\[2\sqrt{2}a\]
\[\sqrt{2}a\]
none of these
Advertisements
Solution
\[4\sqrt{2}a\]

Let OP be the chord.
Let the coordinates of P be \[\left( x_1 , y_1 \right)\]
From the figure, we have:
\[O P^2 = {x_1}^2 + {y_1}^2\] (1)
And,
\[\tan\frac{\pi}{4} = \frac{y_1}{x_1}\]
\[\Rightarrow x_1 = y_1\] (2)
Also,
\[\left( x_1 , y_1 \right)\] lies on the parabola.
∴ \[{y_1}^2 = 4a x_1\] (3)
Using (2) and (3):
\[{x_1}^2 = 4a x_1 \Rightarrow x_1 = 4a\] (4)
∴ From (4), (1) and (2), we have:
\[O P^2 = \left( 4a \right)^2 + \left( 4a \right)^2 = 32 a^2 \]
\[ \Rightarrow OP = 4\sqrt{2}a\]
Therefore, the length of the chord is \[4\sqrt{2}a \text{ units }\]
APPEARS IN
RELATED QUESTIONS
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/36 + y^2/16 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/16 + y^2/9 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/49 + y^2/36 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
`x^2/100 + y^2/400 = 1`
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
36x2 + 4y2 = 144
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
4x2 + 9y2 = 36
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola:
y2 = 8x
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
4x2 + y = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabolas
y2 − 4y − 3x + 1 = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 − 4y + 4x = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
y2 + 4x + 4y − 3 = 0
Find the vertex, focus, axis, directrix and latus-rectum of the following parabola
x2 + y = 6x − 14
For the parabola y2 = 4px find the extremities of a double ordinate of length 8 p. Prove that the lines from the vertex to its extremities are at right angles.
Write the distance between the vertex and focus of the parabola y2 + 6y + 2x + 5 = 0.
Write the coordinates of the vertex of the parabola whose focus is at (−2, 1) and directrix is the line x + y − 3 = 0.
The directrix of the parabola x2 − 4x − 8y + 12 = 0 is
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
x2 + 4y2 − 4x + 24y + 31 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
4x2 + y2 − 8x + 2y + 1 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
3x2 + 4y2 − 12x − 8y + 4 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
4x2 + 16y2 − 24x − 32y − 12 = 0
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
x2 + 4y2 − 2x = 0
Find the equation of an ellipse whose foci are at (± 3, 0) and which passes through (4, 1).
Write the eccentricity of the ellipse 9x2 + 5y2 − 18x − 2y − 16 = 0.
If S and S' are two foci of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and B is an end of the minor axis such that ∆BSS' is equilateral, then write the eccentricity of the ellipse.
If a latus rectum of an ellipse subtends a right angle at the centre of the ellipse, then write the eccentricity of the ellipse.
Find the equation of the ellipse with foci at (± 5, 0) and x = `36/5` as one of the directrices.
The equation of the circle which passes through the point (4, 5) and has its centre at (2, 2) is ______.
The shortest distance from the point (2, –7) to the circle x2 + y2 – 14x – 10y – 151 = 0 is equal to 5.
