Advertisements
Advertisements
Question
Find the equation of the tangent to the ellipse x2 + 4y2 = 9 which are parallel to the line 2x + 3y – 5 = 0.
Advertisements
Solution
We know that the equations of tangents with slope m to the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 are
y = `"m"x ± sqrt("a"^2"m"^2 + "b"^2)` ...(1)
The equation of the ellipse is x2 + 4y2 = 9
∴ `x^2/9 + y^2/((9/4)` = 1
Comparing this with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 9, b2 = `9/4`
Slope of 2x + 3y – 5 = 0 is `-2/3`
The required tangent is parallel to it
∴ its slope = m = `-2/3`
Using (1), the required equations of tangents are
y = `-(2x)/3 ± sqrt(9 xx 4/9 + 9/4)`
∴ y = `-(2x)/3 ± sqrt(25/4)`
∴ y = `-(2x)/3 ± 5/2`
∴ 6y = – 4x ± 15
∴ 4x + 6y = ± 15
APPEARS IN
RELATED QUESTIONS
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrics
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
2x2 + 6y2 = 6
Find the
- lengths of the principal axes.
- co-ordinates of the focii
- equations of directrices
- length of the latus rectum
- distance between focii
- distance between directrices of the ellipse:
3x2 + 4y2 = 1
Find the equation of the ellipse in standard form if eccentricity = `3/8` and distance between its foci = 6
Find the equation of the ellipse in standard form if the distance between directrix is 18 and eccentricity is `1/3`.
Find the equation of the ellipse in standard form if the distance between foci is 6 and the distance between directrix is `50/3`.
Find the equation of the ellipse in standard form if the latus rectum has length of 6 and foci are (±2, 0).
Find the eccentricity of an ellipse if the distance between its directrix is three times the distance between its foci
Show that the product of the lengths of the perpendicular segments drawn from the foci to any tangent line to the ellipse `x^2/25 + y^2/16` = 1 is equal to 16
Determine whether the line `x + 3ysqrt(2)` = 9 is a tangent to the ellipse `x^2/9 + y^2/4` = 1. If so, find the co-ordinates of the pt of contact
Find k, if the line 3x + 4y + k = 0 touches 9x2 + 16y2 = 144
Find the equation of the tangent to the ellipse 4x2 + 7y2 = 28 from the point (3, –2).
Find the equation of the tangent to the ellipse `x^2/25 + y^2/4` = 1 which are parallel to the line x + y + 1 = 0.
Tangents are drawn through a point P to the ellipse 4x2 + 5y2 = 20 having inclinations θ1 and θ2 such that tan θ1 + tan θ2 = 2. Find the equation of the locus of P.
The eccentric angles of two points P and Q the ellipse 4x2 + y2 = 4 differ by `(2pi)/3`. Show that the locus of the point of intersection of the tangents at P and Q is the ellipse 4x2 + y2 = 16
Find the equations of the tangents to the ellipse `x^2/16 + y^2/9` = 1, making equal intercepts on co-ordinate axes
Select the correct option from the given alternatives:
The equation of the ellipse having foci (+4, 0) and eccentricity `1/3` is
Select the correct option from the given alternatives:
If the line 4x − 3y + k = 0 touches the ellipse 5x2 + 9y2 = 45 then the value of k is
Select the correct option from the given alternatives:
The equation of the ellipse is 16x2 + 25y2 = 400. The equations of the tangents making an angle of 180° with the major axis are
Select the correct option from the given alternatives:
The equation of the tangent to the ellipse 4x2 + 9y2 = 36 which is perpendicular to the 3x + 4y = 17 is,
Answer the following:
Find the equation of the tangent to the ellipse x2 + 4y2 = 100 at (8, 3)
Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of `π/2` at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse E: `x^2/a^2 + y^2/b^2` = 1, a2 > b2. If e is the eccentricity of the ellipse E, then the value of `1/e^2` is equal to ______.
On the ellipse `x^2/8 + "y"^2/4` = 1 let P be a point in the second quadrant such that the tangent at P to the ellipse is perpendicular to the line x + 2y = 0. Let S and S' be the foci of the ellipse and e be its eccentricity. If A is the area of the triangle SPS' then, the value of (5 – e2). A is ______.
An ellipse is described by using an endless string which is passed over two pins. If the axes are 6 cm and 4 cm, the necessary length of the string and the distance between the pins respectively in cms, are ______.
Tangents are drawn from a point on the circle x2 + y2 = 25 to the ellipse 9x2 + 16y2 – 144 = 0 then find the angle between the tangents.
The equation of the ellipse with its centre at (1, 2), one focus at (6, 2) and passing through the point (4, 6) is ______.
The points where the normals to the ellipse x2 + 3y2 = 37 are parallel to the line 6x – 5y = 2 are ______.
The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to ______.
Let the eccentricity of an ellipse `x^2/a^2 + y^2/b^2` = 1, a > b, be `1/4`. If this ellipse passes through the point ```(-4sqrt(2/5), 3)`, then a2 + b2 is equal to ______.
If P1 and P2 are two points on the ellipse `x^2/4 + y^2` = 1 at which the tangents are parallel to the chord joining the points (0, 1) and (2, 0), then the distance between P1 and P2 is ______.
The locus of a variable point whose distance from (- 2, 0) is \[\frac{2}{3}\] times its distance from the line \[x=-\frac{9}{2}\], is______.
If the length of the major axis of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\] is three times the length of minor axis, then its eccentricity is______.
Length of latusrectum of the ellipse \[\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,\] is______.
