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Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.?
Verified Answer
Let L1 be a straight line through the origin and L2 be the straight li...
Let the equation of line L1 be y = mx.
Intercepts made by L1 and L2 on the circle will be equal , i. e, L1 and L2 are at same distance from the centre of the circle.
Centre of the given circle is (1/2, -3/2)
therefore,
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Most Upvoted Answer
Let L1 be a straight line through the origin and L2 be the straight li...
To find the intercepts made by the circle on L1, we need to find the points where L1 intersects with the circle.
First, let's find the equation of L1. Since it passes through the origin, its equation can be written as y = mx, where m is the slope of L1.
Next, substitute this equation into the equation of the circle to find the points of intersection:
x^2 + (mx)^2 - x - 3(mx) = 0
Simplifying, we get:
x^2 + m^2x^2 - x - 3mx = 0
Combining like terms, we have:
(1 + m^2)x^2 - (1 + 3m)x = 0
Now, we can find the x-intercepts of this quadratic equation by setting it equal to zero:
(1 + m^2)x^2 - (1 + 3m)x = 0
x((1 + m^2)x - (1 + 3m)) = 0
Either x = 0, or (1 + m^2)x - (1 + 3m) = 0
If x = 0, then y = 0, and this is the origin, which is already given.
If (1 + m^2)x - (1 + 3m) = 0, then x = (1 + 3m)/(1 + m^2)
Substitute this value of x back into the equation of L1 to find the y-coordinate:
y = m * ((1 + 3m)/(1 + m^2))
So, the intercepts made by the circle on L1 are the points ((1 + 3m)/(1 + m^2), m * ((1 + 3m)/(1 + m^2))).
First, let's find the equation of L1. Since it passes through the origin, its equation can be written as y = mx, where m is the slope of L1.
Next, substitute this equation into the equation of the circle to find the points of intersection:
x^2 + (mx)^2 - x - 3(mx) = 0
Simplifying, we get:
x^2 + m^2x^2 - x - 3mx = 0
Combining like terms, we have:
(1 + m^2)x^2 - (1 + 3m)x = 0
Now, we can find the x-intercepts of this quadratic equation by setting it equal to zero:
(1 + m^2)x^2 - (1 + 3m)x = 0
x((1 + m^2)x - (1 + 3m)) = 0
Either x = 0, or (1 + m^2)x - (1 + 3m) = 0
If x = 0, then y = 0, and this is the origin, which is already given.
If (1 + m^2)x - (1 + 3m) = 0, then x = (1 + 3m)/(1 + m^2)
Substitute this value of x back into the equation of L1 to find the y-coordinate:
y = m * ((1 + 3m)/(1 + m^2))
So, the intercepts made by the circle on L1 are the points ((1 + 3m)/(1 + m^2), m * ((1 + 3m)/(1 + m^2))).
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Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.?
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Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.?.
Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let L1 be a straight line through the origin and L2 be the straight line x y= 1 If the intercepts made by the circle x^2 y^2-x 3y=0 on L1 & L2 are equal, then find the equation(s) which represent L1.?.
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