Home
Class 12
MATHS
The tangents at any point P of the circl...

The tangents at any point P of the circle `x^(2)+y^(2)=16` meets the tangents at a fixed point A at T. Point is joined to B, the other end of the diameter, through, A.
The sum of focal distance of any point on the curce is

A

12

B

16

C

20

D

8

Text Solution

Verified by Experts

Tangent at `P(4 cos theta, 4 sin theta) "to x^(2)+y^(2)=16` is `x cos thetay+sin theta=4 " "(1)`
The equation of AP is `y=(sin theta)/(cos theta-1)(x-4)" "(2)`

From (1), the coordinates of the point T are given by
`(4,(4(1-costheta))/(sin theta))`
The equation of BT is `y=(1-cos theta)/(2 sin theta)(x+4)" "(3)`
Let (h,k) be the point of intersection of the lines (2), and (3). Then we have
`k^(2)=-(1)/(2)(h^(2)-16)`
or `(h^(2))/(16)+(y^(2))/(8)=1`
Therefore, the locus of (h,k) is `(x^(2))/(16)+(y^(2))/(8)=1`
Which is an ellipse with eccehntrically `e=1//sqrt(2)`
Sum of focal distance of any points is 2a=8
Considering the circle `x^(2)+y^(2)=a^(2)`, we find the that the eccentricity of the ellipse is `1sqrt(2)` which is contant and does not change by changing the radius of the circle
Promotional Banner

Similar Questions

Explore conceptually related problems

The tagents at any point P of the circle x^(2)+y^(2)=16 meets the tangents at a fixed point A at T. Point is joined to B, the other end of the diameter, through, A. Which of the following does not change by changing the radius of the circle ?

The tangent at any point P on the circle x^2+y^2=4 meets the coordinate axes at Aa n dB . Then find the locus of the midpoint of A Bdot

Find the sum of the focal distances of any point on the ellipse 9x^2+16 y^2=144.

If the tangent at the point on the circle x^2+y^2+6x +6y=2 meets the straight ine 5x -2y+6 =0 at a point Q on the y- axis then the length of PQ is

The tangent at any point on the ellipse 16x^(2)+25y^(2) = 400 meets the tangents at the ends of the major axis at T_(1) and T_(2) . The circle on T_(1)T_(2) as diameter passes through

The tangent at any point of a circle is perpendicular to the radius through the point of contact.

Draw a tangent at any point R on the circle of radius 3.4 cm and centre at P ?

the tangent at any point P on the ellipe x^(2)/6 + y^(2)/3 = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis. The CN CT = _____

The tangent at any point on the curve x=acos^3theta,y=asin^3theta meets the axes in Pa n dQ . Prove that the locus of the midpoint of P Q is a circle.

A tangent at a point on the circle x^2+y^2=a^2 intersects a concentric circle C at two points Pa n dQ . The tangents to the circle X at Pa n dQ meet at a point on the circle x^2+y^2=b^2dot Then the equation of the circle is