Advertisements
Advertisements
प्रश्न
Find the equations of the tangents to the ellipse `x^2/16 + y^2/9` = 1, making equal intercepts on co-ordinate axes
Advertisements
उत्तर
The equation of the ellipse is `x^2/16 + y^2/9` = 1
Comparing with `x^2/"a"^2 + y^2/"b"^2` = 1, we get,
a2 = 16, b2 = 9
Let the required tangent make intercept c on each axis
∴ its equation is `x/"c" + y/"c"` = 1
∴ x + y = c ...(I)
∴ y = – x + c
We know that y = mx + c is a tangent to the ellipse
`x^2/"a"^2 + y^2/"b"^2` = 1 if c2 = a2m2 + b2
Here, a2 = 16, b2 = 9, m = – 1
∴ c2 = 16 × 1 + 9 = 25
∴ c = ± 5
∴ Using (I), the required equations of tangents are x + y = ± 5.
Alternate Method:
The equation of the ellipse is `x^2/16 + y^2/9` = 1
Comparing this with `x^2/"a"^2 + y^2/"b"^2` = 1, we get,
a2 = 16, b2 = 9
The equation of tangents to the ellipse `x^2/"a"^2 + y^2/"b"^2` = 1 with slope m are
y = `"m"x ± sqrt("a"^2"m"^2 + "b"^2)`
The tangent makes equal intercepts on the coordinate axes, its slope = m = – 1
∴ `sqrt("a"^2"m"^2 + "b"^2) = sqrt(16(-1)^2 + 9)` = 5
∴ the equations of required tangents are y = –x ± 5
∴ x + y = ± 5
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 the distance between directrix is 18 and eccentricity is `1/3`.
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)`.
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
Show that the line 8y + x = 17 touches the ellipse x2 + 4y2 = 17. Find the point 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 x2 + 4y2 = 9 which are parallel to the line 2x + 3y – 5 = 0.
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.
Find the equation of the locus of a point the tangents form which to the ellipse 3x2 + 5y2 = 15 are at right angles
Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse
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:
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:
Centre of the ellipse 9x2 + 5y2 − 36x − 50y − 164 = 0 is at
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 ______.
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 ______.
Let the ellipse `x^2/a^2 + y^2/b^2` = 1 has latus sectum equal 8 units – if the ellipse passes through `(sqrt(5), 4)` Then The radius of the directive circle is ______.
The points where the normals to the ellipse x2 + 3y2 = 37 are parallel to the line 6x – 5y = 2 are ______.
The normal to the ellipse `x^2/a^2 + y^2/b^2` = 1 at a point P(x1, y1) on it, meets the x-axis in G. PN is perpendicular to OX, where O is origin. Then value of ℓ(OG)/ℓ(ON) 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 ______.
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______.
The distance between the foci of the ellipse \[x=3\text{cos}\theta,y=4\text{sin}\theta\] is______.
Eccentricity of the conic \[16x^2+7y^2=112\] is______.
