Advertisements
Advertisements
Question
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`.
Advertisements
Solution
Given equation of the ellipse is `x^2/"a"^2 + y^2/"b"^2` = 1
θ1 and θ2 are the eccentric angles of tangent.
∴ Equation of tangent at point P is
`x/"a"costheta_1 + y/"b"sintheta_1` = 1 ...(i)
∴ Equation of tangent at point Q is
`x/"a"costheta_2 + y/"b"sintheta_2` = 1
θ1 + θ2 = `π/2` ...[Given]
∴ θ2 = `pi/2 - theta_1`
`x/"a"cos(pi/2 - theta_1) + y/"b"sin(pi/2 - theta_1)` = 1
∴ `x/"a"sintheta_1 + y/"b"costheta_1` = 1 ...(ii)
From (i) and (ii), we get
`x/"a"costheta_1 + y/"b"sintheta_1 = x/"a"sintheta_1 + y/"b"costheta_1`
Let M(x1, y1) be the point of intersection of the tangents.
∴ `x_1/"a" costheta_1 + y_1/"b" sintheta_1 = x_1/"a" sintheta_1 + y_1/"b"costheta_1`
∴ `x_1/"a"(costheta_1 - sintheta_1) = y_1/"b"(costheta_1 - sintheta_1)` ...(iii)
If cos θ1 – sin θ1 = 0,
cos θ1 = sin θ1
∴ tan θ1 = 1
∴ θ1 = `pi/4`
Since θ1 + θ2 = `pi/2`, θ2 = `pi/2 - pi/4 = pi/4`
i.e., points P and Q coincide, which is not possible, as P and Q are two different points.
∴ cos θ1 – sin θ1 ≠ 0
Dividing equation (iii) by (cos θ1 – sin θ1), we get
`x_1/"a" = y_1/"b"`
∴ bx1 – ay1 = 0
∴ bx – ay = 0, which is the required equation of locus of point M.
APPEARS IN
RELATED QUESTIONS
Answer the following:
Find the
- lengths of the principal axes
- co-ordinates of the foci
- equations of directrices
- length of the latus rectum
- distance between foci
- distance between directrices of the ellipse:
`x^2/25 + y^2/9` = 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 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 equation of the ellipse in standard form if passing through the points (−3, 1) and (2, −2)
Find the equation of the ellipse in standard form if the dist. between its directrix is 10 and which passes through `(-sqrt(5), 2)`.
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
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
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.
Find the equation of the locus of a point the tangents form which to the ellipse 3x2 + 5y2 = 15 are at right angles
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
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:
If the line 4x − 3y + k = 0 touches the ellipse 5x2 + 9y2 = 45 then the value of k is
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 ______.
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 ______.
The tangent and the normal at a point P on an ellipse `x^2/a^2 + y^2/b^2` = 1 meet its major axis in T and T' so that TT' = a then e2cos2θ + cosθ (where e is the eccentricity of the ellipse) 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 ______.
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 ______.
Eccentricity of ellipse `x^2/a^2 + y^2/b^2` = 1, if it passes through point (9, 5) and (12, 4) is ______.
Equation of the ellipse whose axes are along the coordinate axes, vertices are (± 5, 0) and foci at (± 4, 0) 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 ______.
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______.
\[\frac{x^{2}}{r^{2}-r-6}+\frac{y^{2}}{r^{2}-6r+5}=1\] will represent the ellipse, if r lies in the interval______.
Eccentricity of the conic \[16x^2+7y^2=112\] is______.
The distance between the foci of the ellipse \[3x^2+4y^2=48\] is______.
For the ellipse \[3x^2+4y^2=12,\] the length of latus rectum is______.
