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A normal is drawn to the parabola y2 = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P is
- a)4
- b)15/4
- c)6
- d)17/4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A normal is drawn to the parabola y2 = 9x at the point P(4, 6), S bein...
Required intercept will be equal to the perpendicular distance from the focus on the tangent at P.
Tangent at P,
⇒ 12y = 9x + 36
⇒ 9x - 12y + 36 = 0
Tangent at P,
⇒ 12y = 9x + 36
⇒ 9x - 12y + 36 = 0
Most Upvoted Answer
A normal is drawn to the parabola y2 = 9x at the point P(4, 6), S bein...
Problem:
A normal is drawn to the parabola y² = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P is
Solution:
Let's solve the problem step by step.
Step 1: Find the equation of the normal at point P:
To find the equation of the normal at point P(4, 6), we need the slope of the tangent at that point. The slope of the tangent can be found by differentiating the equation of the parabola.
Given equation: y² = 9x
Differentiating both sides with respect to x:
2yy' = 9
At point P(4, 6), substitute the values of x and y into the equation:
2(6)y' = 9
12y' = 9
y' = 9/12
y' = 3/4
The slope of the tangent at point P is 3/4. Since the normal is perpendicular to the tangent, the slope of the normal is the negative reciprocal of the slope of the tangent.
Slope of the normal = -1/(3/4) = -4/3
Using the point-slope form of a line, we can write the equation of the normal as:
(y - 6) = (-4/3)(x - 4)
Simplifying the equation:
3(y - 6) = -4(x - 4)
3y - 18 = -4x + 16
3y = -4x + 34
4x + 3y = 34
So, the equation of the normal at P(4, 6) is 4x + 3y = 34.
Step 2: Find the coordinates of the point of intersection of the normal and the circle:
Since the circle is described on the focal distance of point P as diameter, its center will lie on the normal at point P. Therefore, substituting the x-coordinate of point P into the equation of the normal will give us the x-coordinate of the center of the circle.
Substituting x = 4 into the equation 4x + 3y = 34:
4(4) + 3y = 34
16 + 3y = 34
3y = 34 - 16
3y = 18
y = 6
So, the x-coordinate of the center of the circle is 4 and the y-coordinate is 6. Therefore, the center of the circle is C(4, 6).
Step 3: Find the radius of the circle:
The radius of the circle is the distance between the center C(4, 6) and the point P(4, 6). We can use the distance formula to find the radius.
Distance formula:
√((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the coordinates of the center C(4, 6) and the point P(4, 6):
radius = √((4 - 4)² + (6 - 6)²)
radius =
A normal is drawn to the parabola y² = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P is
Solution:
Let's solve the problem step by step.
Step 1: Find the equation of the normal at point P:
To find the equation of the normal at point P(4, 6), we need the slope of the tangent at that point. The slope of the tangent can be found by differentiating the equation of the parabola.
Given equation: y² = 9x
Differentiating both sides with respect to x:
2yy' = 9
At point P(4, 6), substitute the values of x and y into the equation:
2(6)y' = 9
12y' = 9
y' = 9/12
y' = 3/4
The slope of the tangent at point P is 3/4. Since the normal is perpendicular to the tangent, the slope of the normal is the negative reciprocal of the slope of the tangent.
Slope of the normal = -1/(3/4) = -4/3
Using the point-slope form of a line, we can write the equation of the normal as:
(y - 6) = (-4/3)(x - 4)
Simplifying the equation:
3(y - 6) = -4(x - 4)
3y - 18 = -4x + 16
3y = -4x + 34
4x + 3y = 34
So, the equation of the normal at P(4, 6) is 4x + 3y = 34.
Step 2: Find the coordinates of the point of intersection of the normal and the circle:
Since the circle is described on the focal distance of point P as diameter, its center will lie on the normal at point P. Therefore, substituting the x-coordinate of point P into the equation of the normal will give us the x-coordinate of the center of the circle.
Substituting x = 4 into the equation 4x + 3y = 34:
4(4) + 3y = 34
16 + 3y = 34
3y = 34 - 16
3y = 18
y = 6
So, the x-coordinate of the center of the circle is 4 and the y-coordinate is 6. Therefore, the center of the circle is C(4, 6).
Step 3: Find the radius of the circle:
The radius of the circle is the distance between the center C(4, 6) and the point P(4, 6). We can use the distance formula to find the radius.
Distance formula:
√((x₂ - x₁)² + (y₂ - y₁)²)
Substituting the coordinates of the center C(4, 6) and the point P(4, 6):
radius = √((4 - 4)² + (6 - 6)²)
radius =
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A normal is drawn to the parabola y2 = 9x at the point P(4, 6), S being the focus, a circle is described on the focal distance of the point P as diameter. The length of the intercept made by the circle on the normal at P isa)4b)15/4c)6d)17/4Correct answer is option 'B'. Can you explain this answer?
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