Home
Class 12
MATHS
One extremity of a focal chord of y^2 = ...

One extremity of a focal chord of `y^2 = 16x` is `A(1,4)`. Then the length of the focal chord at A is

A

22

B

25

C

24

D

20

Text Solution

Verified by Experts

The correct Answer is:
B
(i) First find the focus of given parabola
(ii) Then, find the slope of the focal chord by using `m = (y_(2) - y_(1))/(x_(2) - x_(1))`
(iii) Now, find the length of the focal chord by the using the formula `4 a sec^(2) alpha`.
Equation of given parabola is `y^(2) = 16x` its formula is (4,0) since, slope of line pasing through `(x_(1), y_(1))` and `(x_(2), y_(2))` is given by `m = tan theta = (y_(2) - y_(1))/(x_(2) - x_(2))`
`:.` Slope of focal chord having one end point is (1,4) is
`m = tan alpha = (4 -0)/(1 - 4) = - (4)/(3)`
Where `'alpha'` is the inclination of focal with X-axis.
since the length of focal chord `= 4a cosec^(2) alpha`
`:.` The required length of the focal chord
`= 16 [1 + cot^(2) alpha] [:' a = 4 "and" cosec^(2) alpha = 1 + cot^(2) alpha]`
`= 16 [1 + (9)/(16)] = 25` units `[:' cot alpha = (1)/(tan alpha) = - (3)/(4)]`
Promotional Banner

Similar Questions

Explore conceptually related problems

If the length of focal chord of y^2=4a x is l , then find the angle between the axis of the parabola and the focal chord.

AB is a chord of x^2 + y^2 = 4 and P(1, 1) trisects AB. Then the length of the chord AB is (a) 1.5 units (c) 2.5 units (b) 2 units (d) 3 units

If t_(1) and t_(2) are the extremities of any focal chord of y^(2) = 4ax " then " t_(1)t_(2) is _____

If the point P(4, -2) is the one end of the focal chord PQ of the parabola y^(2)=x, then the slope of the tangent at Q, is

If alphaa n dbeta are the eccentric angles of the extremities of a focal chord of an ellipse, then prove that the eccentricity of the ellipse is (sinalpha+sinbeta)/("sin"(alpha+beta))

If (2,-8) is at an end of a focal chord of the parabola y^2=32 x , then find the other end of the chord.

The focal chord to y^2=16 x is tangent to (x-6)^2+y^2=2. Then the possible value of the slope of this chord is (a) {-1,1} (b) {-2,2} (c) {-2,1/2} (d) {2,-1/2}

If aa n dc are the lengths of segments of any focal chord of the parabola y^2=2b x ,(b >0), then the roots of the equation a x^2+b x+c=0 are (a)real and distinct (b) real and equal (c)imaginary (d) none of these

If the mid-point of a chord of the ellipse (x^2)/(16)+(y^2)/(25)=1 (0, 3), then length of the chord is (1) (32)/5 (2) 16 (3) 4/5 (4)12 (5) 32

The locus of the circumcenter of a variable triangle having sides the y-axis, y=2, and lx+my=1, where (1,m) lies on the parabola y^(2)=4x , is a curve C. The length of the smallest chord of this C is