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A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these?
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A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles pas...
Problem: A(1,2) and B(4,-4) are two points on the parabola y^2=4x. Circles passing through the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1=0 B) x^2=0 C) x^3=0 D) None of these.
Solution:
Step 1: Find the equations of the tangents to the parabola at C and D.
Let C and D have coordinates (h,k) on the parabola y^2=4x. Since C and D lie on circles passing through the parabola, the centers of these circles lie on the perpendicular bisector of CD. Thus, the perpendicular bisector of CD passes through the midpoint of CD, which is ((h+1)/2,(k+2)/2). The slope of the tangent to the parabola at C is 1/(dy/dx)|C = 1/(2k/4h)|C = 2h/k, and the slope of the tangent to the parabola at D is 2h/k. Thus, the equations of the tangents to the parabola at C and D are:
(y-k) = (2h/k)(x-h) and (y-k) = (2h/k)(x-h)
Step 2: Find the equations of the circles passing through C and D and the parabola.
Since C and D lie on the parabola, we know that (h,k) satisfies the equation y^2=4x. Thus, we can write k^2=4h, or h=k^2/4. Since the circles passing through C and D also pass through the point (h,k), the equations of the circles are:
(x-h)^2 + (y-k)^2 = (h+2-k)^2 and (x-h)^2 + (y-k)^2 = (h+4+k)^2
Substituting h=k^2/4, we get:
(x-k^2/4)^2 + (y-k)^2 = (k^2/4+2-k)^2 and (x-k^2/4)^2 + (y-k)^2 = (k^2/4+4+k)^2
Simplifying, we get:
x^2 - (k^2/2)x + (k^4/16 + k^2 - 4k - 4) + y^2 - 2ky + k^2 - 4k - 4 = 0 and x^2 - (k^2/2)x + (k^4/16 + k^2 + 4k + 16) + y^2 - 2ky + k^2 + 4k + 16 = 0
Step 3: Find the intersection of the tangents to the circles.
Substituting the equations of the tangents to the circles, we get:
(y-k) = (2h/k)(x-h) and (y-k) = (-2h/k)(x-h)
Solving for x and y, we get:
x
Solution:
Step 1: Find the equations of the tangents to the parabola at C and D.
Let C and D have coordinates (h,k) on the parabola y^2=4x. Since C and D lie on circles passing through the parabola, the centers of these circles lie on the perpendicular bisector of CD. Thus, the perpendicular bisector of CD passes through the midpoint of CD, which is ((h+1)/2,(k+2)/2). The slope of the tangent to the parabola at C is 1/(dy/dx)|C = 1/(2k/4h)|C = 2h/k, and the slope of the tangent to the parabola at D is 2h/k. Thus, the equations of the tangents to the parabola at C and D are:
(y-k) = (2h/k)(x-h) and (y-k) = (2h/k)(x-h)
Step 2: Find the equations of the circles passing through C and D and the parabola.
Since C and D lie on the parabola, we know that (h,k) satisfies the equation y^2=4x. Thus, we can write k^2=4h, or h=k^2/4. Since the circles passing through C and D also pass through the point (h,k), the equations of the circles are:
(x-h)^2 + (y-k)^2 = (h+2-k)^2 and (x-h)^2 + (y-k)^2 = (h+4+k)^2
Substituting h=k^2/4, we get:
(x-k^2/4)^2 + (y-k)^2 = (k^2/4+2-k)^2 and (x-k^2/4)^2 + (y-k)^2 = (k^2/4+4+k)^2
Simplifying, we get:
x^2 - (k^2/2)x + (k^4/16 + k^2 - 4k - 4) + y^2 - 2ky + k^2 - 4k - 4 = 0 and x^2 - (k^2/2)x + (k^4/16 + k^2 + 4k + 16) + y^2 - 2ky + k^2 + 4k + 16 = 0
Step 3: Find the intersection of the tangents to the circles.
Substituting the equations of the tangents to the circles, we get:
(y-k) = (2h/k)(x-h) and (y-k) = (-2h/k)(x-h)
Solving for x and y, we get:
x
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A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these?
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A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these?.
A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A(1,2) and B(4,-4) are two points on the parabola y^2=4x . Circles passing Through A , B intersect the parabola again at C and D. The locus of the intersection of the tangents to the parabola at C and D is A) x-1 =0 B) x 2=0 C) x 3=0 D) None of these?.
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