Advertisements
Advertisements
प्रश्न
Find the equation of the tangent to the ellipse 4x2 + 7y2 = 28 from the point (3, –2).
Advertisements
उत्तर
Given equation of the ellipse is 4x2 + 7y2 = 28
∴ `x^2/7 + y^2/4` = 1
Comparing this equation with `x^2/"a"^2 + y^2/"b"^2` = 1, we get
a2 = 7 and b2 = 4
Equations of tangents to the ellipse
`x^2/"a"^2 + y^2/"b"^2` = 1 having slope m are
y = `"m"x ± sqrt("a"^2"m"^2 + "b"^2)`
Since (3, –2) lies on both the tangents,
–2 = `3"m" ± sqrt(7"m"^2 + 4)`
∴ –2 – 3m = `± sqrt(7"m"^2 + 4)`
Squaring both the sides, we get
9m2 + 12m + 4 = 7m2 + 4
∴ 2m2 + 12m = 0
∴ 2m(m + 6) = 0
∴ m = 0 or m = – 6
∴ These are the slopes of the required tangents.
∴ By slope point form y – y1 = m(x – x1), the equations of the tangents are
y + 2 = 0(x – 3) and y + 2 = –6(x – 3)
∴ y + 2 = 0 and y + 2 = –6x + 18
∴ y + 2 = 0 and 6x + y – 16 = 0.
APPEARS IN
संबंधित प्रश्न
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 minor axis is 16 and eccentricity is `1/3`.
Find the equation of the ellipse in standard form if the dist. between its directrix is 10 and which passes through `(-sqrt(5), 2)`.
Find the equation of the ellipse in standard form if eccentricity is `2/3` and passes through `(2, −5/3)`.
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
A tangent having slope `–1/2` to the ellipse 3x2 + 4y2 = 12 intersects the X and Y axes in the points A and B respectively. If O is the origin, find the area of the triangle
Show that the line x – y = 5 is a tangent to the ellipse 9x2 + 16y2 = 144. Find the point of contact
Show that the line 8y + x = 17 touches the ellipse x2 + 4y2 = 17. Find the point of contact
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 the equation of the tangent to the ellipse `x^2/5 + y^2/4` = 1 passing through the point (2, –2)
Find the equation of the tangent to the ellipse x2 + 4y2 = 20, ⊥ to the line 4x + 3y = 7.
P and Q are two points on the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 with eccentric angles θ1 and θ2. Find the equation of the locus of the point of intersection of the tangents at P and Q if θ1 + θ2 = `π/2`.
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:
If `"P"(pi/4)` is any point on he ellipse 9x2 + 25y2 = 225. S and S1 are its foci then SP.S1P =
Select the correct option from the given alternatives:
The equation of the ellipse having eccentricity `sqrt(3)/2` and passing through (− 8, 3) is
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,
Select the correct option from the given alternatives:
Centre of the ellipse 9x2 + 5y2 − 36x − 50y − 164 = 0 is at
Find the equation of the ellipse in standard form if the length of major axis 10 and the distance between foci is 8
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 ______.
If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a2 is equal to ______.
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 ______.
The eccentricity, foci and the length of the latus rectum of the ellipse x2 + 4y2 + 8y – 2x + 1 = 0 are respectively equal to ______.
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 point on the ellipse x2 + 2y2 = 6 closest to the line x + y = 7 is (a, b). The value of (a + b) will be ______.
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 ______.
A focus of an ellipse is at the origin. The directrix is the line x = 4 and the eccentricity is `1/2`. Then the length of the semi-major axis is ______.
