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Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Then
- a)SP = 2√5
- b)SQ:QP = ((√5+1):2)
- c)the x-intercept of the normal to the parabola at P is 6
- d)the slope of the tangent to the circle at Q is 1/2
Correct answer is option 'A,C,D'. Can you explain this answer?
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Verified Answer
Let P be the point on the parabola y2 = 4x which is at the shortest di...
Equation of normal of parabola is
y + tx = 2t + t3
Normal passes through S(2, 8)
8 + 2t = 2t + t3
t = 2
y + tx = 2t + t3
Normal passes through S(2, 8)
8 + 2t = 2t + t3
t = 2Hence P ≡ (4,4) and SQ = eadius = 2
Most Upvoted Answer
Let P be the point on the parabola y2 = 4x which is at the shortest di...
Shortest distance between a point on a parabola and the center of a circle
- The shortest distance between a point P on the parabola y^2 = 4x and the center S of the circle x^2 + y^2 – 4x – 16y + 64 = 0 is a normal line from the point P to the circle.
- Let the coordinates of P be (h, 2√h) as it lies on the parabola equation.
- The center of the circle is (2, -8) obtained by completing the square on the given circle equation.
- The equation of the normal to the parabola at P is y = -x/√h + 4h + 2√h.
- To find the x-intercept of the normal, substitute y = 0 in the equation to get x = 6.
- Therefore, the x-intercept of the normal to the parabola at P is 6.
Internal division of line segment SP
- The point Q divides the line segment SP internally in the ratio of the squares of the slopes of SP with respect to the circle x^2 + y^2 – 4x – 16y + 64 = 0.
- The distance SP is given by 2√h which is calculated by substituting the coordinates of P in the distance formula.
- Hence, SP = 2√5.
Slope of the tangent to the circle at Q
- The slope of the tangent to the circle at Q is the negative reciprocal of the slope of the line joining the center of the circle to the point Q.
- The slope of line SQ is given by (2√h + 8)/(h - 2) by using the two-point formula.
- Therefore, the slope of the tangent to the circle at Q is 1/2.
Therefore, the correct options are A, C, and D.
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Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Thena)SP = 2√5b)SQ:QP = ((√5+1):2)c)the x-intercept of the normal to the parabola at P is 6d)the slope of the tangent to the circle at Q is 1/2Correct answer is option 'A,C,D'. Can you explain this answer?
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Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Thena)SP = 2√5b)SQ:QP = ((√5+1):2)c)the x-intercept of the normal to the parabola at P is 6d)the slope of the tangent to the circle at Q is 1/2Correct answer is option 'A,C,D'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Thena)SP = 2√5b)SQ:QP = ((√5+1):2)c)the x-intercept of the normal to the parabola at P is 6d)the slope of the tangent to the circle at Q is 1/2Correct answer is option 'A,C,D'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let P be the point on the parabola y2 = 4x which is at the shortest distance from the center S of the circle x2 + y2 – 4x – 16y + 64 = 0. Let Q be the point on the circle dividing the line segment SP internally. Thena)SP = 2√5b)SQ:QP = ((√5+1):2)c)the x-intercept of the normal to the parabola at P is 6d)the slope of the tangent to the circle at Q is 1/2Correct answer is option 'A,C,D'. Can you explain this answer?.
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